I like to win that game.
The use of the present tense here suggests a habit. Perhaps you play poker and you always like to win.
You: I play poker every Saturday.
Me: Do you mind if you lose?
You: Yes! I like to win that game!
You're explaining to me that you enjoy/prefer to win when you play poker. In this context, "game" = an enjoyable diversion (cards, sport etc)
I would like to have won that game.
This refers to a specific game that has taken place in the past, and which you lost. You're telling me that you wish you had won. In this context, "game" = a specific competitive interlude. A game of football. A game of tennis. A game of cards.
What is difference in meaning meaning between them?