What is the difference between the two?

Other than wording, the meanings are identical, although the first could be taken as a general observation or comment whereas the second suggests what he is doing now.

"Kasparov plays chess." -- This says that he is able to and regularly does play the game.

"Kasparov is playing chess." -- This says that he is playing at this moment, or at the moment mentioned elsewhere in the sentence or passage.

Kasparov plays chess makes it sound like a hobby or fact. Kasparov is playing chess sounds like they are playing it in that moment.

To is a preposition which begins a prepositional phrase or an infinitive. Too is an adverb meaning "excessively" or "also." Two is a number. Many other words in English which reflect the number two are spelled with tw: twin, twice, between, tweezers, etc.

"...playing chess" means he is doing that right now or preparing to do so if it has never been done before.

"...plays chess" means he has done so already.