Here's my list.
Pros:
1. Sneak archery. I find any other type of sneaking extremely hard. Taking people out with daggers will be almost impossible until you have a very high level sneak. Archery sneaking is very easy, on the contrary.
2. Poison usage. I invested a bit in alchemy just so I could make some poison. It's usually useless, but hey, it's fun to see a guy go mad and attack his team mates, or fall to the ground paralyzed. There's also a perk in alchemy that allows your poison to be used for two arrows, or two strikes with a melee weapon.
3. Staggering. Once you can get to it, Power Shot is an amazingly useful perk. It'll stop opponents from firing an arrow, using spells, swinging a weapon, or moving towards you for about a second or so. This gives you time to get in another shot, potentially staggering them again. I've taken out so many annoying enemies by slowing them down this way.
4. Adrenaline Rush. That feel when you get an almost impossible shot, or you have to take out several enemies at once so you're trying to figure out a combination of arrows and potions to deal the most damage.
5. Less damage taken. As a gamer, I get annoyed when I see the red on my screen that indicates I got hit. It just annoys me. With range it's easier to avoid damage.
6. Archer to archer battles. It's fun to fight another archer, timing your attacks so you don't get hit by their arrows but you hit them. These are the easiest since they aren't going to try getting close to you.
7. Light armor. As an archer, light armor is advised, and heavy armor isn't necessary. Sure, you could be a tank archer, but that's kind of boring. The agility is nice.
There are probably more, but I can't think of it right now.
Cons:
1. Missing. Especially for beginners, missing shots can be a problem. You eventually get over it and learn how to hit every time though. Getting the eagle eye > steady hand perk will definitely help with this. Even though some people don't like this perk, it's pretty useful if you want to hit all your shots. Especially with rare arrows that you don't want to waste.
2. Getting jammed. At the beginning of the game, archery is harder since melee opponents get up close and you are pretty much forced to change weapons. This gets a lot easier once you get certain perks I'll talk about later.
3. Hitboxes on objects. Oh god. Skyrim has horrendous hitboxes on objects, so if you shoot several inches or even a foot/2 feet above an object, it'll still hit it and get stuck. Extremely annoying if you're trying to take cover while shooting, or if you're shooting someone behind cover. No fix that I know of for this.
4. Slower for getting through dungeons. I find that rushing in kamikaze with a sword will get me the same amount of kills in less time. Sure, I'll take damage, but that's what restoration magic is for. If you want to be an archer, be prepared to spend a lot of time sneaking around corners, rather than shooting stuff. This will probably lessen if you are a strong archer that doesn't need the sneak bonus.
5. Feeling guilty when losing good arrows. Not really a big deal, but when I have a strong arrow that was expensive, I hate missing with it.
Those are all the cons I can think of. There are probably more. However, I think archery is one of the coolest ways to play the game. Especially with a bow that has a fire enchantment
Most useful perks IMO:
1. Overdraw.
2. Power shot.
3. Eagle Eye/Steady Hand (level 1, 2 is excessive).
4. RANGER. So useful for backing up.
5. Quick shot.
I would write more, but I'm tired =3