This is a simple question with a slightly complicated answer. The simple response is that Marine pilots are in the Marine Corps and Navy pilots are in the Navy, but that's not necessarily helpful to you. Warning: long post to follow.
-Both attend the same schools to winging, with Marines taking a 6 month detour to The Basic School prior to showing up to Pensacola. Depending on the needs of the services, Marine or Navy requirements may be stricter to pass different phases of flight school. For example, during the height of the Iraq/Afghanistan Wars, the Marines were apparently taking everybody with a pulse and pushing them through, but when I was in flight school Marines were required to get higher scores than Navy students throughout (to include higher grade requirements for jets, etc.).
During flight school, the playing field is pretty level. There are a mix of Navy/Marine students and IPs (plus Coasties) and generally every phase is relatively "service blind." Marine and Navy students will often be paired together for flights and so on. There are some subtle cultural differences between Marine/Navy studs but for the most part everybody gets along. Post selection and wings at the end of flight school is where the difference really starts.
For life in the fleet: I will lead with I am biased and have a certain opinion because I am a Marine. While there are things I definitely don't like about my job, I think I'm happier than I would be in the Navy. My Navy friends hear me talk about the Marine Corps and Thank God they went Navy. So, your mileage may vary.
The simplest, and not most accurate, way to think about this is that Marine aviation exists to support ground Marines. Naval Aviation exists to support the fleet and its strategic objectives. So for example while the Marine Corps and Navy both fly helicopters, for example, Marine helicopters train to shooting missiles/rockets/guns in support of ground forces as their primary mission while Navy helicopters train to antisubmarine or surface ship warfare as their big focus.
Marines will generally be collocated or close to the ground units they support, while Navy units will be dis-aggregated across the coast at helicopter or jet bases and come together as an air wing for deployment.
Who gets the most flight time ebbs and flows. I',m pretty sure I get more (and more interesting) hours than my Navy helicopter counterparts, we both get more than most of the guys flying jets in either service, and everybody gets less than the people who fly the big multi-engine aircraft. The "Marines get the worst stuff" is a bit of a misnomer these days. We definitely have struggles with readiness and budget, but the tide is turning and the aircraft in most communities are pretty new with a few exceptions.
The biggest cultural difference in my mind is that in the Navy each of the big communities is considered relatively equal. Everybody knows that everybody else has an important part to play, whether they drive a submarine, a surface ship, or a helicopter, and separate career paths are considered valid. People in the Navy seem to understand better that different communities require different things and it is expected of you to become a tactical expert in fighting your aircraft/ship/whatever over your knowledge of how the whole Navy operates.
In the Marine Corps, people outside the AirWing don't necessarily "get it" and aviation can become a kind of second-class community among officers. Marine Air exists to support the grunt and we love doing it but that comes with the knowledge that our needs are not necessarily understood by the higher headquarters. The non-flying job burden is generally higher in the Marine Corps than in the Navy and you are expected to be a jack of all trades and master of everything. Your quality of life will be lower. But, ultimately, I find it incredibly rewarding. When I deployed, I knew that if we went to combat (we didn't) I wouldn't be shooting for some anonymous dude on the radio, it would be to support the guy who lives across the hall from me. I love leading Marines and think my helicopter mechanics are some of the finest, hardest working Americans out there.
Both will have their pain outside of the cockpit after an initial flying billet of 3-5 years, or maybe more. In the Navy, that comes with tours to the Carrier Staff. In the Marine Corps, that normally comes as a tour as a Forward Air Controller (FAC) or Air Officer with a grunt unit, which is generally considered a rewarding, if difficult, tour.
As an Academy grad I obviously had a choice between Navy and Marine Air and I went Marine with no regrets but it's not for everyone.
Best,
gun (wo)man Capt Hurricane