I'm going to answer your question in two parts, the first being why Mantine sucks in general and the second being what makes Skarmory and other bulky Flying-types oh so much better.
So, why does fluffy little Mantine suck so badly?
Well, there are a few reasons, the first being awful defensive typing. Water/Flying has a weakness to two common attacking types, Electric and Rock. On the defensive side, he has resistances to Fire, Fighting, Steel, Water and Bug.
But some of those are really common attacking types?
True, but Mantine can't do squat back to the attackers due to its really, really poor movepool and offensive stats. Coverage-wise, his best attacks are Earthquake, Rock Slide, Acrobatics, Waterfall, Seed Bomb, Gunk shot, Bullet Seed… which would be awesome if his Attack stat wasn't an abyssmal 40. His Sp. Attack is slightly more useful at 80, but his Special movepool is limited to Air Slash, Scald, Hydro Pump, Signal Beam, Ice Beam and Hidden Power. As for his support movepool, he has a good one for Doubles; Tailwind, Wide Guard, and Helping Hand. Unfortunately, there are other Flying types who do that better.
So what makes Mantine so inferior to all the other Flying-types?
Let's compare them, shall we?
Epic Pokémon Battles of Historyyyyy!!
Every other Flying-type Pokémon!
Skarmory has vastly superior defensive typing and defensive stats, as well as support/wall movepool.
Talonflame steals the show with priority Tailwind, Brave Bird and Roost recovery, as well as better offensive typing.
Zapdos has far more relative bulk than Mantine. He can go Bulky Tailwind, speedy offense or even Toxic stall (not recommended).
Togekiss is sooo much better in both Singles and Doubles due to being bulkier, having a faster Tailwind, Serene Grace, a much better support movepool, better defensive typing and Follow Me redirection.
Gyarados has the exact same typing as Mantine, but better HP and Defence. He also has the coveted Intimidate ability, as well as Moxie for sweeping. He's faster, stronger, and has a much better movepool. He can run bulky, support, Dragon Dance, Assault Vest, and has a Mega Evolution. Gyarados is probably the number one reason Mantine is never used.
In conclusion, Mantine is vastly inferior to virtually every flying-type due to his poor stat distribution, movepool, and typing. But don't fret; Mantine will fly through the ocean providing a home for orphaned Remoraid for eons to come.
>No mention of Steve Irwin anywhere.
You all should be disappointed in yourselves. ~ DarkTyphlosion
Mantine is also modelled after a stingray, which killed Steve Irwin the Crocodile Hunter, who was my entire childhood. So yeah.