There is really no argument between the two.
Lapras has much better stats, with a 535 total while Azumarill has 410. Lapras has a better movepool and typing. It's abilities are better in competitive, Lapras has a better position in the metagame.
>Although the transition to BW introduced many strong threats, Azumarill continues to be a force to be reckoned with. Its ability Huge Power gives it a monstrous Attack stat comparable to that of behemoths such as Groudon. While it is rather slow, it has access to STAB priority in Aqua Jet, allowing it to strike first even when it is slower. Likewise, it is quite bulky, possessing 100 / 80 / 80 defenses; with the HP stat Azumarill is able to make 101 HP Substitutes. In a tier filled with Registeel and Dusclops, being able to set up Substitutes that survive Seismic Toss is a rather big deal. Unfortunately, Azumarill is really slow and rather one-dimensional, making it very predictable. Nonetheless, whether it's Azumarill's brute force, bulk, STAB priority, or dastardly good looks that earns it a spot on your team, you can rest assured knowing that it will be an effective contributor to your success.
>Lapras has lain in the depths of NU since DPP, and it's a mystery why. Being one of the few specially defensive Water-types in the tier, Lapras has a niche in countering Shell Smash Gorebyss with ease. Lapras's secondary Ice typing is somewhat of a letdown, but don't let this put you off; STAB Ice Beam coming from a base 85 Special Attack is nothing short of scary. Unfortunately, Lapras is cursed with a Stealth Rock weakness and no way to heal itself outside of Rest and Water Absorb, but thanks to its decent typing it has many opportunities to switch in. Lapras has a rather large movepool consisting of oddities such as Dragon Dance, Perish Song, and even Curse.** If it wasn't weak to Stealth Rock and its Speed was a bit higher, Lapras would be a top-tier threat, but for now it'll stay with its NU friends.
For lapras, I like JCM's set:
Lapras (F) @ Damp Rock
EVs: 252 Def / 216 SDef / 32 HP / 8 Spd
Calm Nature (+SDef,-Atk)
- Rain Dance
- Perish Song
Azumarill (M) @ Choice Band
Trait: Sap Sipper
EVs: 252 HP / 252 Atk / 4 Def
Adamant Nature (+Atk, -SAtk)
- Aqua Jet
- Body Slam
- Ice Punch
Lapras in rain, Azumarill in general. Lapras IS better, but Azumarill has a different niche. You can't compare the two THAT much.