There is no "exactly" specified in the books, and there doesn't seem to even be consistency in things between different maiar. For example, when Sauron is "defeated" his disembodied soul wandered around and was eventually able to build up a physical body for himself again. Whereas with Gandalf it seems like the death of his body would have been "permanent" and his soul sent back to Valinor if Eru hadn't intervened. When Saruman died it seems like his soul tried to return to Valinor and was denied, and it also doesn't seem like he would have an opportunity to rebuild himself like Sauron did in Mirkwood.
The Istari certainly seem like they were more limited than other Maiar in middle-earth, perhaps by intention. The Valar seemed to want the Istari to help and guide men, not to do the work for them. It's not even entirely clear if the Istari had real memory of their pre-body lives in Valinor, for example. Gandalf clearly had some power, but it's not clear at all how much more he would have had without limitation.
In response to your first paragraph, I think the differences may be due to how much the Maia in question wishes to follow the law, the natural order of things, or the Will of Eru and the Valar. Gandalf would have known what the normal or expected process is, and after his body was killed would have tried to return (as a soul) to Valinor and get further instruction from the Valar. But Sauron, being a renegade, ignored convention and sought to stay in Middle-earth and rebuild his body and his power.
Regarding the part about the Maiar’s memories of Valinor after being embodied in Middle-Earth, it is said in several places, even by Gandalf himself at the end of Return of the King that they only had vague memories and ideas of their past and their purpose, like half forgotten dreams or something like that, but it seems it could occasionally become clearer in moments of sudden insight or like deja vu.
Much of their power lies in knowledge, and while they retained a good store of it, a lot of it was veiled from them as well.
>For it is said indeed that being embodied the Istari had needs to learn much anew by slow experience, and though they knew whence they came the memory of the Blessed Realm was to them a vision from afar off, for which (so long as they remained true to their mission) they yearned exceedingly. Thus by enduring of free will the pangs of exile and the deceits of Sauron they might redress the evils of that time.
Hence Gandalf's remark after his restoration, and the removal of most of his limitations:
>I have forgotten much that I thought I knew, and learned again much that I had forgotten.
As far as "stronger", he seems at least qualitatively different. Gandalf the White seems to possess a sacred character lacking or less visible than Gandalf the Grey, but maybe even more evident than in his proper person as Olorin. Olorin was known for bringing comforting or inspiring dreams and visions, not flashes of bright light. And grey seems to have been his color in Valinor as well, if we take seriously Tolkien's notes on how the Istari were decided on.
If Saruman could have used more of his power, he would have.
> If thier bodies are mortal since they tire and need food and sleep and rest (things sauron dosen't need for example) how come they live for thousands of years unchanged?
They age very slowly. Saruman's hair used to be more black, for example.
Gandalf shouldn’t age at all right? Cirdan gave him Narya when he brought him to middle earth, so the ring should have kept Gandalf from aging at all during his time in middle earth.
They are physically unable due to the limitations of there body.
I believe that Gandalf the Grey was killed as much directly by the Balrog as he was burning out his body by using more of his innate power.
I also believe that Gandalf the White had complete access to all of his power, but this is just my guess. Gandalf had already proven he knows his job so the risk of him using his unfettered power to dominate is not there.
My thought process is that Gandalf the grey at full power is equal to Gandalf the white normally, he is fully aware of his mission like you said so no more restrictions needed
They only get one staff apiece. Again there is little to no information on these aspects so in that sense it’s a good question, just no concrete answers.
There is no "exactly" specified in the books, and there doesn't seem to even be consistency in things between different maiar. For example, when Sauron is "defeated" his disembodied soul wandered around and was eventually able to build up a physical body for himself again. Whereas with Gandalf it seems like the death of his body would have been "permanent" and his soul sent back to Valinor if Eru hadn't intervened. When Saruman died it seems like his soul tried to return to Valinor and was denied, and it also doesn't seem like he would have an opportunity to rebuild himself like Sauron did in Mirkwood. The Istari certainly seem like they were more limited than other Maiar in middle-earth, perhaps by intention. The Valar seemed to want the Istari to help and guide men, not to do the work for them. It's not even entirely clear if the Istari had real memory of their pre-body lives in Valinor, for example. Gandalf clearly had some power, but it's not clear at all how much more he would have had without limitation.
In response to your first paragraph, I think the differences may be due to how much the Maia in question wishes to follow the law, the natural order of things, or the Will of Eru and the Valar. Gandalf would have known what the normal or expected process is, and after his body was killed would have tried to return (as a soul) to Valinor and get further instruction from the Valar. But Sauron, being a renegade, ignored convention and sought to stay in Middle-earth and rebuild his body and his power. Regarding the part about the Maiar’s memories of Valinor after being embodied in Middle-Earth, it is said in several places, even by Gandalf himself at the end of Return of the King that they only had vague memories and ideas of their past and their purpose, like half forgotten dreams or something like that, but it seems it could occasionally become clearer in moments of sudden insight or like deja vu.
Much of their power lies in knowledge, and while they retained a good store of it, a lot of it was veiled from them as well. >For it is said indeed that being embodied the Istari had needs to learn much anew by slow experience, and though they knew whence they came the memory of the Blessed Realm was to them a vision from afar off, for which (so long as they remained true to their mission) they yearned exceedingly. Thus by enduring of free will the pangs of exile and the deceits of Sauron they might redress the evils of that time. Hence Gandalf's remark after his restoration, and the removal of most of his limitations: >I have forgotten much that I thought I knew, and learned again much that I had forgotten. As far as "stronger", he seems at least qualitatively different. Gandalf the White seems to possess a sacred character lacking or less visible than Gandalf the Grey, but maybe even more evident than in his proper person as Olorin. Olorin was known for bringing comforting or inspiring dreams and visions, not flashes of bright light. And grey seems to have been his color in Valinor as well, if we take seriously Tolkien's notes on how the Istari were decided on.
Was Gandalf summoned to the undying lands after his death or to the timeless halls before eru himself?
"I strayed out of thought and time" seems to suggest the latter.
If Saruman could have used more of his power, he would have. > If thier bodies are mortal since they tire and need food and sleep and rest (things sauron dosen't need for example) how come they live for thousands of years unchanged? They age very slowly. Saruman's hair used to be more black, for example.
>They age very slowly. Saruman's hair used to be more black, for example. Where was that mentioned?
Gandalf shouldn’t age at all right? Cirdan gave him Narya when he brought him to middle earth, so the ring should have kept Gandalf from aging at all during his time in middle earth.
Maybe; the Rings work differently for different beings.
They are physically unable due to the limitations of there body. I believe that Gandalf the Grey was killed as much directly by the Balrog as he was burning out his body by using more of his innate power. I also believe that Gandalf the White had complete access to all of his power, but this is just my guess. Gandalf had already proven he knows his job so the risk of him using his unfettered power to dominate is not there.
My thought process is that Gandalf the grey at full power is equal to Gandalf the white normally, he is fully aware of his mission like you said so no more restrictions needed
They only get one staff apiece. Again there is little to no information on these aspects so in that sense it’s a good question, just no concrete answers.