Different pronunciations of "Pakistan" versus "Afghanistan"?

No, their actual (native) pronunciations have a similar final syllable. Roughly, PAH-ki-STAHN and uf-GHAH-ni-STAHN, where "AH" stands for the vowel of 'a' in "father". (The 't' is also closer to dental, as in "pasta".
(IPA: /pɑːkistɑːn/ and /ɐfɣɑːnistɑːn/, but these are the IPA vowels I use for Indian languages, and may not be perfect.)

The Language Log had a post on this; the comments may have some helpful discussion.

They should be pronounced the same, the -stan suffix means "land of", "home of" or "nation of".

So: Pakistan = Land of the Paki's Afghanistan = Land of the Afghani's

The same applies for Uzbekistan, Turkmenistan, and any number of other countries in that region of the world.

I suspect that Obama pronounces Afghanistan differently because people wouldn't know where he was talking about if he said it in a way that doesn't conform with how the US media has been pronouncing it. It's probably this same reason that the Iraq is pronounced eye-rak rather than ee-rak which is closer to the Arabic, people might think they are talking about another country.

In AmE, all the a's in 'Afghanistan' and 'Pakistan' are the same as that in 'cat', that is, the 'a' in '-stan' should be pronounced the same.

As noted, Obama may have experience with Pakistanis and their own pronunciation, which probably has a different pronunciation of the 'a'.