Here are some key differences. Even though I'm telling you this on an online forum, I'd advise ignoring such resources altogether for the UK fellowship process - there's tons of misinformation floating around, and most of what is said on sites like this is colored by the fact that people with all sorts of different credentials and backgrounds are looking at all sorts of different schools. Your school should have a fellowship adviser who can help, or your career services office might be able to put you in touch with a previous recipient. Same thing applies for law school: conventional wisdom on TLS often doesn't mean much when it comes to individual candidates.
To win any one of these, you'll need professors who have your back, great non-academic recommendations, and some combination of awards, intensive activities, jobs, esoteric academic interests, etc. that you can describe in a coherent manner (i.e. in a way that implies you are diverse in your interests and talents, yet focused on something for the future). In a sense, it's all a game. The winners are usually no smarter or more interesting than the ones who just missed the final cut. The way I see it is this: you shouldn't plan your life around winning one of these or going to HYS. These are crapshoots. Just do your best and get involved in things you're passionate about, inside and outside the classroom.
Rhodes: Oxford only, 2-3 years. Funded by Rhodes trust, with application process divided by state. You'll need your university's endorsement. Stereotype: very leadership oriented, and somewhat less focused on academic ability and intentions. The athletics requirement can be fulfilled by basically anything - do you like to take long walks? Dance? This requirement is a bit outdated and very few recipients I know are great athletes!
Marshall: Basically any UK institution, 2-3 years. Funded by the UK gov't, and administered through the consulates. You'll also need your university's endorsement for this one. Stereotype: more academic, with a slight tendency to favor people who plan to go to non-Oxbridge/LSE universities. Allows you to do two one-year programs at different universities, if you want.
Gates: Organized a lot like the Rhodes, but for Cambridge, 1-5 years. Favors people committed to public service. No endorsement from your university - basically your department at Cambridge has to rank you as its top candidate for you to be considered. Ridiculously well-funded.
UK Fulbright: Very tough to get, because there are so few. You need your school's endorsement, and unless you are the cream of the crop it's best to apply to non-Oxbridge/LSE schools if you're going for this.
Mitchell: For graduate study in Ireland or Northern Ireland. Also extremely difficult because of the small number of seats. I don't know much about this, but it's often mentioned in the same breath as the above scholarships.
Incidentally, these are not the only ways to get funding for UK degrees - there are plenty of other scholarships out there.
I won't be back on TLS for the foreseeable future - I've exhausted all the procrastination time I can allot at this point
Good luck with everything!