The biggest difference between Hinduism and Buddhism

[Blue3]

These two religions are very similar, having influenced each other over thousands of years. But there are also differences, and this seems to be the biggest one (correct me if I misunderstand):

Q: Who are you, really and truly? Are "you" your body? No, it changes, and most of your cells are replaced every 7 years. Is your true self your mind/personality? No, your personality is different from year to year, even minute to minute. Your personality at 95, 50, 20, and 7 years old are probably very different. So who are you, what is your true self, when you strip everything else away?

Buddhism's Answer: You're nothing. When all of those temporary conditions are stripped away, there's nothing left. Fully realizing this, and being at peace at with this truth, brings serenity and enlightenment, and allows you to escape beyond life's suffering. Give up your desires and attachments.

Hinduism's Answer: You're the Ultimate Reality. When all of those temporary conditions are stripped away, only your atman (personal soul) is left. And your atman is the same as everyone's atman, and is even in fact identified with Brahman (ultimate reality). What you truly crave is infinite being, infinite consciousness, and infinite bliss. But in reality, you are atman, and atman is Brahman... you already have everything you truly want! Just fully realize this truth, it will bring you happiness and enlightenment, and allow you to be liberated from life's cycles of suffering. You already have everything that you truly desire.

Correct?

[Blue3]

Hinduism being polytheistic and Buddhism being non-theistic seems more basic.

Hinduism isn't really polytheistic. It's more panentheism/monism. They choose whichever god or idea gives them the best connection to the Ultimate Reality, whether it's through bhakti yoga or jnana yoga or karma yoga or raja yoga.

[Blue3]

I know very little about this subject, but the Hinduism part seems right except for the "you already have what you need and all you need to do is realize that" part. I thought that "the point" of Hinduism was to progressively improve/purify the condition of your atman by adhering to the Dharma, and, once your atman is fully "pure," to escape reincarnation and be one with Brahman? Am I just totally wrong here?

Yes. I know Hinduism much more than Buddhism.
http://en.wikipedia.org/wiki/Satcitananda
http://en.wikipedia.org/wiki/Advaita_Vedanta#Identity_of_Atman_and_Brahman

You already have what you really want. But you can only escape reincarnation after fully realizing it and adapting accordingly.

[Blue3]

The thing I've never quite understood about Hinduism and Buddhism is why one should want to end the cycle of reincarnation.  Frankly, if reincarnation were true, experiencing a variety of lives sounds much more preferable, even if some particular incarnations are not preferable.

In the Hindu view, it's perfectly acceptable to live as many lives as you want. They just think that eventually, you'll see that an infinite number of lives without enlightenment would never satisfy what you truly want (and the "good news" in Hinduism is that you already have what you truly want, you just need to fully realize it). Liberation (moksha) is just one of the 4 paths in Hinduism, but all paths are considered valid... the other three being duty (dharma), worldly success (artha), and pleasure (kama, NOT karma). Also, most Hindus and Buddhists don't literally believe in reincarnation, and see it more as a metaphor for how we are really all the same after you strip away all the superficial characteristics of body and personality.

[Blue3]

Any other thoughts on the fundamental differences between the 3rd and 4th largest religions?