Yes the biggest problem in Peer to Peer multiplayer is keeping game state synchronized between all clients. So you need active mechanizm which makes sure that game states are synchonized between all clients. This of course means that the slowest computer with worst connection will cause the most lag to all other players.
But the biggest advantage of Peer to Peer multiplayer is that you don't require dedicated server for it (no aditional expesnes for dedicated servers) becouse anyone can host a game at any time.
Another advantage is that using Peer to Peer you can distribute processing between different clients. This in the end alows using of more complex game mechanics.
Unfortunately this introduces huge chances for players to cheat with modified clients.

But if you wanna make a multiplayer game where noone could cheat all I can say is good luck. Why? There is no easy solution to this.
For instance I remember a cheat mod for Far Cry which alowed you to easily shot at other players. All you needed was to have another player in sight and mod automatically calculated needed trajectory for bulet. This means you didn't even have to aim and was still making perfect kills.
When PunkBuster service got implemented they prevented this mod to work until it was updated to work even with PunkBuster protection.

So the only way to prevent players from cheating in multiplayer games is making games so that client simply sends player imputs to server and server simpy returs visual and audio information back to player. There is absolutely no processing made on client. So you need extreemly powerfull servers for this. But a viable solution is actually moving server into cloud to get enough processing power.
And since all processing would be made on server side this would also alow you to play extremly demmanding games even on crappy computer (no need for advanced graphics cards, no need for large amount of RAM, no need for large HDD space as you would require only tiny application on your computer). All it is necessary is for that computer to be able to smotly play any streamed movie with desired resolution.
But for this to work you need excelent network connection with minimal lag.