What I think the Metroid series does better than the Zelda series (or any other series, for that matter) is exploration. It's just you wandering alone (the vast majority of the time) in a hostile world and discovering secrets along the way. What makes it better is that, in order to find all the secrets, you have to master the controls; just look at the GBA entries and the devilish tricks you have to pull off to get some of the later missile expansions and health tanks. Also, there are no obvious indications that many of the rooms even exist; you really have to explore the entire landscape instead of looking for key clues like the Zelda series relies on (far too often IMO; if it can somehow balance modern Zelda views and the first NES game's views I'll be happy).
Also, taken as a whole I like boss (and mini-boss) battles in Metroid better than Zelda. Most of the time they are more engaging and, well, more of a threat to me. Zelda generally does better when its big baddies are roughly human-sized (Demise, Ganondorf, Ghirahim, Dark Link, Scervo, Darknut, etc.) or when they are monstrous but have several methods of attack and/or can be brought down in a variety of ways (Odolwa, Twinmold, Bongo Bongo, etc.). But Metroid bosses as a whole are much more aggressive and require much quicker thinking and acting to take down. Their gigantic bosses don't suffer the same fate Zelda bosses do (the bigger they are, paradoxically the easier they are to take down due to their easily-identifiable weak spot) and they deal out some heavy damage (Quadraxis, Thardus, Emperor Ing, and Metroid Prime are my personal favorites). And several of the smaller enemies will really give you a run for your money (Dark Samus, Boost Ball and Spider Ball guardians, Gandrayda, Rundus, Mecha Ridley, and Nightmare are the ones that kicked my butt the most).
Still, I like the Zelda series better than the Metroid series; it has a much higher awesome-to-crap ratio (9-1) than Metroid (6-3), disregarding the simply okay games. And while I love both the universes they're set in, the Zelda one slightly edges out over Metroid. It has the distinct advantage of weaving a grand tale that sprawls centuries and timelines, whereas Metroid is more laser-focused on Samus and her endeavors, with the ancient Chozo race thrown in for good measure.
As for what I think of the individual games: Skyward Sword, Ocarina of Time, and Majora's Mask are my absolute favorite games ever, and most of the other games in the series rank up very highly as well, with the only real misstep in the series being Adventure of Link. Metroid Prime is... perfect. No joke, I can't think of a single flaw in the game. If it weren't for the aforementioned 3 Zelda titles it would be my absolute favorite game. Prime 2 is really damn good as well, and I very much enjoyed Prime 3, Zero Mission, Fusion, and Super Metroid. However, Prime Hunters, Pinball, and most of all Other M just don't fit with the rest of the series IMO (even if on a gameplay perspective they're pretty good). I haven't played Metroid 2, it being on the GameBoy, so I don't know what to think of it (although I'm waiting patiently for AM2R to be completed!)