Just as a cursory comparison, a couple things:
Buddhism tends to be about "self" as an illusion. Existence (at least this earthly, material one) is inherently one of attachments which bring suffering. The effort is one of removing attachment, including that to the idea of a self. I think one can leave Buddhism (especially Zen) to a psychological or philosophical level, without necessarily involving spiritual ideas like an afterlife or soul, etc. More traditional or religious Buddhist belief tends to be one of reincarnation, one's karma and attachment keeps them in the cycle of being, and enlightenment is to detach oneself from this cycle and attain nirvana.
While Hinduism predates Buddhism and arguably is the predecessor in many of Buddhist beliefs, there are some interesting differences. There is certainly more of a spiritual cosmology in Hinduism, generally pantheistic. Although, the concept of Brahman is kind of like God the Father, while Atman might be considered similar to the Holy Spirit in some sense. Whereas Buddhism tends to leave a Void in the practice of detachment, Hinduism and the Atman considers there is a Self behind all selves. Instead of a Void there's a fundamental Unity.
You should definitely look into Alan Watts. (I will share anything I find that's relevant, when I get a chance.)
I'm no scholar, and still learning, but I do feel a calling of sorts to these philosophies/religions. I've read a smattering of sources, but ones I recommend, so far: "Living Buddha, Living Christ" by Thich Nhat Hanh and "Training the Mind and Cultivating Loving-Kindness" by Chogyam Trungpa.
Thrall to the Wire of Self-Excited Circuit.