They are pretty similar in stats, except Quagsire is more physical, and Gastrodon is more special. With Gastrodon having beefier stats than Quagsire, meaning it has more Hp and better stats in general.
Gastrodon has the Storm Drain ability which is good for double battles, and Quagsire has Water Absorb, which is pretty good, but it's very reliant on having your opponent make the mistake of hitting you with a water move in the first place.
Quagsire, I would think has a better movepool in general, they both have access to recover, making them pretty good at walling especially with their typing. Howeve, Gastrodon may be better in that regard with its Hp. Both of their movepools are very similar, however, and I would say Gastrodon, with its better stats and all, wins over Quagsire, especially in double battles.
I am not putting movesets for them here, because I would rather have you figure it out for yourself, if that's alright. Hope I helped.