It really depends on the game and the particular implementation in that game (what people call "netcode").
For example, like Hackworth said above, certain games like Garrysmod or TF2 wouldn't be possible with P2P, because the server's particular configuration is part of the game (like it has mods, or plays a particular set of maps).
In other games, like Titanfall for example, you actually want server configurations to be identical, and because they have a smaller number of players per game, the difference between the host player and any other player is minimal. P2P also uses matchmaking, so getting into a game is pretty quick, the matchmaking server just puts you on a list, and then makes games out of people on that list. No digging through server menus or finding out that your favourite few servers are all at capacity.
Now, dedicated server systems can have matchmaking (usually called "quick join" or something), but many don't, and even those that do are often not as good as in a P2P based game (like it will stick you in a server that is bad for some reason, like has only 2 people playing or only plays one map or something).
TLDR; if server configuration is important, like with mods, or the servers support a large amount of players, like 32 vs 32 TF2 servers or any MMO ever, dedicated servers are absolutely superior to a P2P implementation. However, in cases where the network traffic is relatively low, and server configuration doesn't matter/is identical, P2P has some advantages over dedicated servers, and relatively few downsides in terms of lag.