Why are the old Pokémon TCG sets not allowed in modern tournaments?
There are a good collection of Pokemon TCG tournament rules and resources here - you may find some of it useful.

Looking at the rules for tournament formats, I see that there is a format called "Modified" that restricts players to using more recent sets - but there is also an "Unlimited" format where any cards ever printed are fair game.  So, in theory, there's no reason why you couldn't find an Unlimited tournament to play your older cards in.  Lobby your local tournament organizers to run one, if they aren't doing so already!

Why would people prefer to run Modified tournaments?  Well, as Aramis correctly states above, the manufacturers have a vested interest in people buying new cards in order to stay competitive.  However, it's not all for evil, economic reasons.  It would be deeply unfair to new players if, every time they turned up at a tournament, they were crushed by long-time players with vast collections of powerful, out-of-print cards.  It's not necessary for the old cards to be stronger than the new cards for there to be a problem: the players who have been collecting for a decade will have so many more options to choose from that by definition they will be able to build better decks.  And if an older, out-of-print card is actively unbalanced, then it's very unfair to ask new players to spend big bucks getting hold of these, just to compete.

Compare and contrast to Magic the Gathering: there is a format called "Vintage" which is quite popular - but to build a truly competitive Vintage deck you need cards like Black Lotus and Moxes, which cost hundreds of dollars apiece!  If all Magic tournaments were Vintage, then only players who have been playing for 20 years or have tens of thousands of dollars to spend would stand a chance of winning a game.  And nobody wants that!