Why does Amazon care? Does Texas not, like other states, charge a use tax which would cause Amazon's Texas customers to pay the same amount as if sales tax were applied?

Nobody pays the use tax, because nobody's heard of it, and nobody makes you pay it. Also, Texas doesn't have an income tax, only property tax, so Texas wouldn't have a chance of enforcing it short of major invasions of privacy.

States really are drooling at the prospect of collecting sales tax from online vendors.

Here's to hoping Amazon can continue to fight them off.