Ezreal is one of those champions that can be built several ways. As pure AD, as an AP caster, or as a hybrid.
All of Ezreal's skills scale off of AP. Essense Flux(0.7 AP ratio) and Archane Shift(0.75 AP ratio) both scale solely on ability power ratios while his Mystic Shot(1.0 AD ratio + 0.2 AP ratio) and Trueshot Barrage(1.0 AD ratio and 0.9 AP ratio) scale off of both AD and AP ratios.
Advantages include:
- AP Ezreal will have higher burst damage. This will allow Ezreal to play a more Guerilla play style where he comes in, harasses with his skills and then back off. He does not need to stay in range for his auto-attack which is the main source of damage for any AD carry.
- His Mystic Shot can proc on-hit effects so building a Lichbane will hit like a truck. this again leads to very strong harass.
- His ultimate Trueshot Barrage is a global ultimate and can deal A LOT of damage if it hits. An AP Ezreal's Trueshot Barrage will be much stronger than an AD Ezreals since AP it is cheaper to build than AD.
Disadvantages include:
- Less sustained damage because AP Ezreals will be limited by their cooldown and mana.
- Much more dependant on the Summoner's ability to use skill shots because every single one of Ezreal's skill is a skill shot. Missing these skill shots essentially means you can't deal any damage.
- Unable to utilize Ezreal's passive Rising Spell Force and Essence Flux's attack speed boost very well. An AP Ezreal's attack damage would be too low to benefit from attack speed much.
- Unable to utilize Mystic Shot's on-hit effect fully. With the exception of Sheen or Lichbane, an AP Ezreal will not be building any items that will proc effects while attacking (Mystic Shot counts as physical attack). So life steal or slow benefits that an AD Ezreal will benefit from will not be available to an AP Ezreal.
With these advantages and disadvantages in mind, whether you choose to go AP or AD on your Ezreal is entirely dependant on your play style.
Let's look at each item, and how it affects Gangplank:
Philosopher's Stone - this is a strong item on many champions. It gives him the mana regen he needs to use Parrrley, HP regen, and GP/5.
Wriggle's Lantern - provides armor, AD, life steal, a unique passive and a free ward. This is another strong pick, especially if jungling Gangplank.
Mercury Treads - these boots are considered the default tank boots, due to MR and tenacity.
Spirit Visage - provides health, magic resist, CDR, and a boost to healing. I don't recommend this item because it doesn't synergize well with any of his abilities.
Frozen Heart - provides armor, mana, CDR, and an aura debuff. This item is best vs an auto-attack heavy team, otherwise I would skip it.
Thornmail - provides armor, and deals magic damage when auto-attacked. I don't recommend this item on Gangplank, because a clever opponent will stop focusing Gangplank when they see you've purchased it. Since your goal is to be a tank, this is counter-productive.
Warmogs Armor - provides the most health in any item, and good health regen. Helpful for staying alive.
Atma's Impaler - provides 15 AD for every 1k HP, armor, and crit chance. A solid pick if you have extra health from runes/items.
The first build you've listed is very defensive. You end with 229 armor and 55 MR. This build would be most effective against a team that relies on auto-attacks (Master Yi, Ashe, etc). It's going to be weaker vs a team doing primarily magic damage. Warmog's and Atma's provides more health (good if you have armor/MR), and some damage, which helps
If you're jungling Gangplank, it's common to rush a Wriggle's first. If you're top lane,
you're going to focus on spamming Parrrley instead, so I'd recommend rushing a Philosopher's stone. Sheen is a common pick, because Parrrley trigger's Sheen's bonus damage. It also builds into a Triforce, an item that boosts all of Gangplank's abilities.
Every game is different, and you have to build vs your enemy team. Of all the items listed, Mercury Treads and a Philosopher's stone are always viable. Wriggle's is very common to see on him, and most of the time will be a good pick. From there, build a sheen if you're using Parrrley a lot, or health, armor, and/or magic resist depending on what's doing the most damage to you.
tl;dr: Mercury Treads, Philosopher's Stone, and Wriggle's Lantern are almost always good picks. Everything else is situational.
Best Answer
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.
Escaping Ganks
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
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.