This is something I've been thinking about for awhile, and I was wondering if anyone here would be able to help me!
Specific examples I was thinking of would be the P-47 Thunderbolt and the F4U Corsair, or the P80 Shooting Star and the F9F Panther, however this is something I've noticed to be fairly farther reaching than just these four. Why were these planes, which were developed roughly around the same time and used for the same purpose, so different? Is it solely because one was meant to launch from a Carrier? Was that one restriction really drastic enough to warrant a completely different design? Was political influence a major factor?
I know that today we're designing a single fighter, the F-35, to be used by both services with respective variants. Why wasn't this approach used back then? Wouldn't having one design to maintain make logistics easier?
Operating from a carrier has different requirements than operating from a land base, especially in the 1930's-1940's before advances in carrier and aircraft technology. An aircraft operating from a carrier needed to be able to:
Be sturdy enough to stand being slammed repeatedly into the carrier deck for landing, and then having a strong enough rear fuselage to contain the force of the arrestor hook. This over-engineering costs weight, which is why, as a rule of thumb, carrier aircraft of the war performed worse than equivalent land-based aircraft. For example, the naval version of the Spitfire, the Seafire, needed reinforcing strips in the rear fuselage and had a reduction in ceiling and speed.
In World War 2 with non-angled flight decks, carrier aircraft needed a low enough stall speed to be able to land safely on a deck. Therefore, wings with higher lift and thus more drag were typically preferred.
Carrier aircraft often needed some sort of space reduction mechanism, such as wing folding, in order to fit inside carrier hangars. These mechanisms are heavy and also need to be over-engineered to stay rigid when desired.
Of course, those are general rules. There were several instances of naval aircraft outperforming land based ones both during the war and after, such as the Zero. Today, there is less distinction between land and carrier aircraft because a) multi-role fighters can do all sorts of jobs rather than having dedicated bombers and fighters, b) angled flight decks and more powerful catapults allow larger and faster aircraft to operate from carrier decks, and c) modern jet fighters are often strong enough to handle carrier landings whether they are designed for it or not, because of the higher speeds and weights they operate at anyways. Also the higher thrust-weight ratio of today's aircraft means that relatively lower amounts of lift don't matter much for take-off.
As to your specific examples: the P-47 Thunderbolt and the F4U Corsair were not used nor designed for the same purpose. The Thunderbolt was designed to be a high altitude interceptor, operating from land bases. The Corsair was intended to operate from carriers. Thus the Thunderbolt had to be large and heavy to fit its turbocharger. The Thunderbolt was top-of-the line technology when it first came out. The Corsair was, in some respects, primitive: the outer wing panels on all wartime Corsairs were fabric covered. The requirements generally for a carrier based aircraft are far different than a land based one and go beyond merely being stressed for landing and take-off. Space on a carrier is always at a premium, and added maintenance space for complex systems such as turbochargers is discouraged. They are designing the F-35 to be used by all services, but the versions for each service are very different, and it remains to be seen if the various designs will be successful. They tried this before with the TFX in the 1960's and it was a spectacular failure. (The Navy version, at least)