We never play where you can see your opponents cards at any time.

How does seeing the cards actually affect things? Is it basically playing to win then?

If anyone has played with seeing them or keeping them secret, can you tell me the

basic difference of game play?

Well, we usually play to win when we play AGoT.

Playing with open information, as the rules suggest you do, makes it so it's not a memory game and you know what your opponent can play. You can pick the weakest card in your hand that will guarantee you victory. Or if you know you're going to win, you might try to risk playing a slightly weaker card to get it out of your hand, expecting them to throw away a card expecting to lose the battle. It makes a big difference not playing cards blindly.