There's a lot of different cheese cutting implements ... so it's probably better to discuss advantages / disadvantages by type.
Wire mounted to a cutting board : Useful for really soft cheeses like chèvre and brie, but I'm not really a fan as you have to clean out the little groove, and it takes up a lot of room for something with only a few uses (it also works for cutting logs of refrigerated cookie dough). If you're cutting a log of chèvre, you get a similar cut by looping it with dental floss and pulling, and it'll deform the log less. Also works well with medium firm cheeses.
Wire mounted on a handle with a roller : Works on semi-soft to medium firmness cheeses (or cheese food); anything that'll hold its shape in a block (even Velveta), but not what you'd consider a 'hard' cheese. There's less drag than a knife, and you can get consistent thickness slices (some are adjustable, some you just change the angle of the handle relative to the block to get a thinner slice) ... but they have the problem that the roller can get gunked up, and if the block's wider than the cutter, you'll have to take a knife to the block first.
Cheese Plane : (looks kinda spatula-ish, with a slicing slot in it). Works best for medium firm cheeses; easy to clean, you get consistent thickness slices (although, only one thickness), and it can be a little faster than using a knife if you're trying for really thin slices. (if you need thicker slices, you'd have to double or triple up, removing any speed advantage). It actually can be used on something wider than the slot if the cheese isn't too firm. And, in a pinch, it can also double as a vegetable peeler, as I've learned from a friend.
When we start getting to "cheese knife", there's a few different things that the term might be referring to:
- a spreading knife : rounded tip, used for serving soft cheeses, but can still cut firmer stuff.
- knives with a fork on the front; may also have holes through the blade so the cheese will release rather than dragging; used for serving medium firm cheeses.
- a spade : short, wide, pointed tip (actually looks like the spade on a deck of cards) : used for prying off chunks of hard cheeses.
I've seen other ones that look more like chisels; I've never used them. I'd assume they'd be best for medium-firm cheeses that aren't in too large of a block.
Personally, I typically use this cheap 'micro-serated' paring knife that I think I got two in a pack at a dollar store about 15 years ago. It's horrible for cutting just about everything else, but it's amazing for most cheeses; it might be that the teeth aren't really aligned well anymore, but it keeps the cheese from dragging along the blade, so I can get really clean slices quite quickly; it works okay for everything but really hard cheeses (good point, but not wide enough to get a flake off, really soft cheeses (see the dental floss comment above), or really crumbly cheeses.