What are the similarities and differences the extinction of mammoths and the possible extinction of Asian elephants?

This is a difficult question to answer because nobody knows for sure why mammoths became extinct.

However, one hypothesis is that human activity (particularly over-hunting), was responsible for the extinction of mammoths, and Asian elephants are also threatened with extinction largely due to hunting. Another possible cause of the mammoth's extinction was habitat loss because of climate change. Habitat loss also effects Asian elephants, but for them, it is due to deforestation caused by humans.

Note that there is one thing that all extinctions have in common. Extinction occurs when a species is unable to adapt to some change in the conditions of their environment. This may be climate change, habitat loss, the disappearance of their food supply, or even the appearance of a new species that over-hunts them, etc.