Istighaatha means to ask for ghauth which means the lifting or removal of harship. The difference between this and suplication (du'a) is that istighaatha only originates from one in some sort of difficulty looking for relief or help. So, du'a is more general while istighaatha is more specific - i.e., it is a special case of du'a. Put another way, every act of istighaatha is and act of du'a, but not every every act of du'a is necessarily an act of istighaatha.
Allah has ordered us to make du'a to him ALONE and has warned us against supplicating any others along with Him or instead of Him. This meaning is found throughout the Qur'an and the Sunnah and very many places. Allah said:
{And the mosques are surely for Allah so do not supplicate any other along with Allah.} Al-Jinn: 18
The purpose of the mosques is to worship Allah and all other activities relating to that such as collecting sadaqa, distributing it, planning for jihad, for calling others to Islam, etc. A Muslim can never supplicate others along with Allah anywhere and especially not in the mosques which are established solely for His worship.
{So do not suplicate any other diety along with Allah and so become one of the punished.} Ash-Shu'araa: 213
Calling to others along with calling to Allah (let alone calling to others instead of Allah) leads to his punishment in the hereafter. It is shirk and shirk is the one sin which Allah stated in the Qur'an that He will not forgive of anyone. The only way to avoid Allah's punishment is to repent and rid oneself entirely of all forms of shirk.
{Supplicate your Lord in humility and secretly. Surely, He does not love those who go beyond} Al-A'raaf: 55
A Muslim should sincerely and humbly ask Allah for his needs. That should be done either in secret or, if in public, to oneself and not spoken out loud. That is best in order to stay clear of ostentation (showing off). Allah does not love those who go beyond the boundaries of what He has ordered us and commanded us, such as by asking Allah improperly (e.g., "Give me a station equal to that of Muhammad in paradise"), or making du'a in a loud voice or crying it out. Let alone directing one's supplication to others along with Allah!
In the following, Allah points out that any calls to help directed to any other than Him are nothing but useless, pointless words since these creations who are called out to cannot even help themselves, let alone those who call to them.
{...That is Allah your lord. His is the dominion and those to whom you call other than Him do not even own the skin of a date seed. If you call to them, they do not hear your supplication. Even if they heard they would not respond to you. And, on Qiyama they will reject your associating (them in worship with Allah) and no one can inform you as can the All-Knowing.} Faatir: 13-14
In the following, Allah rejects the acts of all who direct their worship to others along with Allah - all of which are creations of Allah and entirely within his power and none of which can harm or benefit those who call to them in any way. They do not even possess the powers and abilities of the living humans who call to them in supplication such as hearing, seeing, etc. Then Allah orders his Prophet (sas) to challenge them to launch any and all schemes against him without mercy or hesitation - in order to make apparent the powerlessness of those to whom they call along with Allah.
{Do they associate that which has created nothing and they themselves are created? And they are unable to help them nor to help themselves? Even if you call them to guidance, they do not follow you. It is the same for you if you call them or if you just keep quiet. Those to whom they call besides Allah are surely slaves just like them. So call them and let them respond to you if you are truthful! Do they have legs with which to walk? Or, do they have hands with which to strike? Or, do they have eyes with which to see? Or, do they have ears with which to hear? Say: Call your partners and scheme against me without any hesitation.} Al-A'raaf: 194-195
There is only One who whom our du'a, our istighaatha and any form of worship and supplication should be directed and that is Allah subhaanahu wa ta'ala. Only one who is capable of creation, capable of causing us harm or bringing us benefit and who controls life, death and resurrection is acceptable to be worshipped! Allah said:
{And they took dieties beside Him. They create nothing and they are created. They do not possesses the ability to harm themselves nor to benefit themselves. They do not possess the ability to bring death nor life nor resurrection.} Al-Furqaan: 3
More amazing than all of that is that some of those to whom supplication and other forms of worship is directed are themselves believers. Allah said:
{Say: Call those who you claim beside Him. They cannot keep any harm from you nor deflect it. Those to whom you call are themselves seeking the means to their Lord - which of them is closest - and they hope for His mercy and they fear His punishment. Surely, the punishment of your Lord is something to be feared!} Al-Israa': 56-57
Abdullah ibn Mas'ood (ra) said: "Some humans used to worship individuals among the Jinn. Those Jinn became Muslim, but those humans persisted in worshipping them. About this, Allah send down: {Those to whom you call are themselves seeking the means to Allah}." Muslim
The Prophet (sas) said: "When you ask, ask Allah and when you seek aid, seek aid from Allah." At-Tirmidhi who rated it 'hassan sahih'.
As for istighaatha, or seeking aid and relief from some adversity, it can never be sought from any created being except in that of which they are capable. When the Jew who was fighting with one of Pharaoh's people asked Musa to come to his aid in the fight, he used the word istighaatha in the following verse:
{... and the one who was from his group sought his aid against the one who was from among his enemies...} Al-Qasas: 15
As for that which is not within the ability of the one being called, it cannot be sought from him. And, that which can only be fulfilled by Allah, can only be sought directly from Allah. So, whoever turns to other than Allah in suplication for something of which only Allah is capable has committed shirk and has associated another with Allah in worship. Ibn Taimia said:
"Du'a is one of the forms of worship. Whoever directs it to a created thing who is dead or absent and sought their aid even though this act was not commanded by Allah nor His Prophet (sas) neither an order of obligation nor an order of allowance has committed innovation (bid'a) in his religion and has committed shirk against the Lord of the worlds and has followed other than the path of the believers." (Majmoo' Fatawa Ibn Taimia 1/312)
First: Supplicating others besides Allah who are dead or otherwise absent and regardless if these are among the prophets of Allah, the most righteous believers, the jinn or any others and turning to them for the meeting of one's needs or relief from some difficulty. This is shirk against Allah. Turning to other besides Allah with one's needs or difficulties is the root of all shirk in the history of mankind. This dead one's actions have come to an end. He can do nothing to help or harm himself, let alone the one who is calling to him. And likewise the absent one. He has nothing to do with the situation and can do nothing for the caller. Allah is the one who is close and the one who alone can bring benefit.
Examples of this include the statement of some of the ignorant lost folk who say: "O Sheikh! Give me aid and help me against my enemy." Even more detestable than that is the one who says to this dead or absent one: "Forgive me and accept my repentance." It has even been heard in some Muslim lands that people have said when the rain came down very heavily: "O my sir Abdul-Qaadir, be easy on your slaves." All of this is major shirk against Allah (shirk akbar) which removes one from the fold of Islam. Its perpetrator - in the context of the Islamic state - is to be informed and given a chance to repent. If he repents, that is good for him and if not, his killing is lawful.
Second: Asking for du'a from the dead or the absent either prophets or righteous individual for example by saying: "Ask Allah for me...". Just as the Christians do with Maryam and others. This is an innovation (bid'a) which was not practiced by anyone from the righteous first generations of this Ummah. No one with any intelligence can doubt the forbiddance of such actions. It is not allowed to face a grave in prayer nor to head to a grave when supplicating Allah Most High. So, asking the dead one for something is even more obviously prohibited. It is not lawful to ask the dead one for anything nor to take one's difficulties to them nor to ask them to ask Allah on one's behalf, though some of that was lawful when he was still alive. The companions of the Prophet (sas) were tested by all kinds of adversity after the passing away of the Prophet (sas) and not one single one of them ever went to the grave of the Prophet (sas) or the grave of anyone else and complained of these difficulties, nor did they ask him (sas) to relieve them, nor did they ask him (sas) to ask Allah for them. This is known of necessity to be bid'a and something not ordained by Islam in any fashion.
Third: Seeking a means to Allah (At-Tawassul) through a righteous individual or by the status of a righteous individual with Allah. In other words, while calling to other than Allah in the apparent, the perpetrator claims that they are actually supplicating Allah and that the name of the righteous man is in there for 'influence' due to that man's high station before Allah. This, too, is an innovation and something not sanctioned by Islam. Again, not a single one among the companions (ra) sought a means to Allah via Muhammad (sas) or via the status of Muhammad (sas) with Allah, and he is surely the best of the righteous ones. They were the best of generations and the most knowledgeable of Islam, of the Prophet (sas) and of the way he taught us. As for the statement of Umar: "O Allah, when we suffered from drought, we used to seek a means to You through our Prophet and we used to receive rain and now we seek a means to you through his uncle..." This was the "seeking of a means" by way of asking a living individual to make du'a, something of which they are capable. So, when the Prophet (sas) was alive, he made the du'a for rain and it was granted. Umar decided that they would ask the Prophet's uncle Al-Abbas to make this du'a after the passing of the Prophet (sas). This is completely different from "seeking a means" to Allah via an individual or via the status of an individual before Allah. In fact, this statement is proof against the prohibited form of tawassul because, had it been lawful, the Umar and the Muslim would surely have used the Prophet (who had passed away) and not turned to Al-Abbas (who was still alive).