When talking about advantages and disadvantages, it is necessary to first set a context. Each champion have advantages and disadvantages in different situations. These are several situations in which an AD carry will have to be good at to become effective as an ranged AD carry.
Early Game Farming/Harrassing
Caitlyn is notorious for her early game harass and farm. Caitlyn has the longest range (with the exception of Tristana once she reaches lvl 12) in the game at 650 and is excellent at poking at her enemy. If she utilizes her passive Headshot correctly, she can put unexpected bursts of damage on her enemy. In addition to that, her Piltover Peacemaker is a long range skillshot that does not require targeting. This results in a very one sided harass from Caitlyn.
Miss Fortune on the other hand has a much shorter range at 550. However her skill Double Up is also very good at harass if you are able to position correctly. This makes her harass much less reliable though. Her skill Make It Rain isn't a very good harass because it deals damage over time only if the enemy is within the area where this skill is casted. The enemy will rarely stand still in this area so it will result in a very negligible amount of damage. Not to mention the fact Make It Rain is a skill that is usually maxed last. However Make It Rain is very good at last hitting multiple minions at once if you find there are several minions that are about to die and you can't possibly simultaneous get all the last hits.
It should be noted that Miss Fortune's passive Struts gives her up to 70 additional movement speed after 7 seconds of not getting hit. This can be utilized to position herself to use Double Up so that bullet bounces to the enemy Champion. The speed boost is also very helpful to dodge skillshots such Caitlyn's Piltover Peacemaker.
Inevitably during a game you will be ganked at one point. Because a ranged AD carry is usually very squishy, having the ability to escape and avoid damage is almost a must on an effective ranged AD carry.
Caitlyn is very effective in this respect. Her Yordle Snap Trap if placed in strategic bushes can warn her of an incoming gank and give her time to fall back to her turret because it provides vision and immobilizes the enemy that sprung the trap for 1.5 seconds. In addition to that, the initial cast of the Yordle Snap Track will very briefly provide vision inside of a bush. This can be very useful to scout whether there are additional enemies in the bush or not.
If somehow the ganker was very sneaky and comes undetected, Caitlyn's 90 Caliber Net is a great escape tool. It shoots a net at the enemy to slow them AND shoots Caitlyn backwards. This in addition to flash(if she has that as a summoner skill) will widen the gap between the ganker and Caitlyn.
Miss Fortune unfortunately lack such utility. Her only skill that can aid in her escape is her Make It Rain which damages and slows enemy champions in a small area. However, note that her Make It Rain can also briefly provide vision during the duration of the skill if cast in a bush. Although to be fair her passive gives Miss Fortune up to 70 bonus movement speed. This is lost however if she is hit even once(this will mostly likely happen during a gank).
Damage Output during 1v1
Although at first it may seem like Caitlyn has all the advantages, it is not quite the case in terms of pure damage output. Because two of Caitlyn's skills are utility skills, she has less burst than Miss Fortune. Miss Fortune has a built in steroid skill called Impure Shots which passively added additional magic damage to her auto attacks for each attack that stacks up to 4 times. When activated she will attack 30-50% faster and cause her enemy to heal only 50%. This is terrific for 1v1 another AD carry who will most likely rely on their life steal to stay alive. In addition, Miss Fortune's Double Up skill will reset her attack timer. This means if she auto attacks once, uses Double Up, she will immediately attack again regardless of what her attack speed is.
Caitlyn on the other hand has an ultimate called Ace in the Hole which is VERY effective at securing a kill from an enemy who is running away with low health. Or for chipping down the enemy's health before they are even close enough to hit you. Caitlyn's Headshot is great for 1v1 especially if the fight takes place in the bushes. She can effectively deal 150% every 3-4 attacks.
Damage Output during team fight
Although Caitlyn's Ace in the Hole is great for single target, it isn't very effective during a team fight. Ace in the Hole can be intercepted by another Champion(usually the tank who will take much less damage). Miss Fortune's ultimate Bullet Time on the other hand is very effective and lends itself great to the play style of a ranged AD carry. She is able to stay way in the back and as soon as the team fight starts, she can use Bullet Time's AOE(area of effect) damage to hit all the enemy. Then after most of them are low, you can come in and take the kills.
Caitlyn's Piltover Peacemaker does do AoE damage in a line, however each consecutive enemy hit by it gets reduced damage so that it really doesn't end up being quite effectively. She is more likely to do more damage with her autoattacks at that point.
Other factors to take in account is the statistics of the Champion without any items. Most of their base statistics are very similar but the main difference is their base health. Miss Fortune will start out with a base of 520 (435 + 85/lvl) health while Caitlyn will only start with 470 (390 + 80/lvl) health. In addition, Miss Fortune will continue getting 5 more health per level than Caitlyn will. This makes Miss Fortune slightly tankier than Caitlyn with respect to base health.