With deep stacks you need a stronger hand to play for all the chips. If you are 300BB deep, even KK can become a fold preflop, or you dont put in the extra raise to keep the pot smaller and see a flop. Like maybe you 3-bet and face a 4-bet, and then you just call that 4-bet to keep your opponents range wider.
If on the other hand you are 10BB deep on the bubble of the tournament, you can usually open shove any two cards from SB, or at least any two but junk. So its not really true, that you play tighter with a short stack. But there is less room to try to hit something or outplay the opponent. So typically you are looking to either play for all the chips or get out early.