-I love having the ability to hug and kiss my girlfriend before, during, and after games. It's otherwise still great fun playing with friends and loved ones, i.e. the people you typically want to play games with the most.
-Cheating is nearly impossible, so you're almost guaranteed a fair game.
-Usually no worries about lag or having a game cut out suddenly. A game that does (that's not online) either has too much going on at once, or has bad design.
-It's easier for the group to get what is preferred, such as a particular map or weapon set to be used during play.
-You don't normally have to pay for extra crap. Maybe an extra controller, if anything, but that's about it. Everything else is available for you automatically.
-You don't get to play with some cool people you otherwise wouldn't normally get a chance to play with.
-Unless you have LAN set up, you can't play with more than four people.
-Sometimes the people you want to play with in local multilayer aren't available at the time you want to play.
-Sometimes, there is extra content available online, such as more maps and items. Extends the life of the game and usually makes it more fun.
-Usually, you can play with up to sixteen people easily. Sometimes more, sometimes less, but either way, you're most likely going to play with more than four people.
-Somebody is always available for play, and sometimes you can meet some good people.
-Most of the people you meet online, however, are assholes in one way or another, but your most common asshole is probably the troll, who just does what he/she does for his own personal lulz (not like it's funny or takes much thought anyway).
-To add to that, the treatment of women is particularly bad, and is otherwise something you normally won't get in local multiplayer. Click the image below to see a good example of the typical bull shit involved with that.
-There are quite a few cheaters online. Although some people are unfairly judged as cheaters, some people do little things that give away the fact that they are indeed cheating somehow, such as being able to see your display name at all times, or if they only need to shoot one or two bullets from a sub-machine gun to kill you.
-If there aren't people cheating, you're bound to find somebody using a cheap tactic, or you'll find somebody who's insanely good (i.e. somebody who spends far too much time playing the game). Either way, getting killed constantly by any of these people ruins the fun for everyone else.
-Don't be surprised if the game ends suddenly, or lags because of a bad connection. Hell, don't be surprised if you can't even get online, either.
-Extra shit usually must be paid for, whether it be a monthly subscription to the online service running your games, or add-ons and stuff like that.
-Certain options are usually predetermined, mostly maps. Sometimes it can be changed with a vote if a particular map is unsatisfactory to most players, but that doesn't happen even half of the time you play. A bad setting is almost guaranteed any time you play.
-Good luck being able to play your games online forever. Servers may be shut down; a lot of games no longer have online multiplayer due to the feature not being supported by the company anymore. Even so, don't expect to play games like Mario Kart 64 or the original Super Smash Bros. online.
So, what would I pick? I think it should be obvious: local multiplayer kicks online's ass any day of the week. I'll play online games if I'm really bored and have nobody around to play a game with me, but otherwise, I'll take giving my girlfriend hugs and kisses over some idiot insulting me for the easiest attempt at "humor" ever any day of the week.