Something to keep in mind is that, with 31 IV's and 252 EV's, Azumarill hits the magic number of 404 HP at level 100, allowing for 101-HP substitutes, which can withstand a level-100 Night Shade or Seismic Toss. While in-game this is rare, due to requiring FR/LG for seismic toss chansey, it's something to keep in mind for emulator play, and to watch for once Poke-Bank is released.
Aside from that, Azumarill getting the Fairy type has transformed it from a huge-power gimmick to a real threat. Belly Drum + Aqua Jet is an interesting, if gimmicky, setup. Fairy-type lets it switch into a dragon-type move for free, and the threat of play-rough usually causes a switch. If they lock themselves into Outrage, it gives you free time to set up for a sweep.
The biggest vulnerability Azumarill has is its reliance on Huge Power to deal damage. Cofagrigus walls Azumarill well, having the bulk to take a hit (usually), getting rid of Huge Power via Mummy, and burning with Will-o-Wisp. Aside from this, it can't do much to actually KO Azumarill, but it keeps it from posing much of a threat.
Other options include Power-Up Punch, for another gimmicky set-up option. Waterfall works for water-type STAB with twice the power of Aqua Jet and 100 accuracy to Play Rough's 90, although the loss of coverage and/or priority is felt deeply.
Aqua Ring is another interesting option, working well with Substitute and Leftovers for a more defensive/stalling set, and allowing it to Sub-Protect stall to some degree (again, another gimmick).
Thick Fat takes away almost all options for a decent offense, but makes it an even better wall against Fire and Ice moves. Then again, it already resists both of these types, so unless you REALLY need to sponge damage from Mega Charizard X's STAB moves, you're almost always better off with Huge Power. If you are using Thick Fat for whatever reason, remember that Azumarill's Special Attack got buffed to base 60. Still not very threatening, and it lacks much of a movepool to use with it, but it's not the worst out there.
Eviolite Marill... Once again relegated to the Gimmick corner. It has better overall defenses, yes, but base 20 is NON-EXISTENT offense. Plus, it's not like the eviolite boost puts it too far ahead of Azumarill, and the HP loss is really felt.
Perhaps the biggest weakness for Azumarill is its limited movepool. Most every set you see will tend to run the same moves, meaning it's easy to predict what's coming out.
Overall, Azumarill is probably one of the greatest beneficiaries of the Fairy Type in 6th Gen, except perhaps Mawile. While most of its sets rely on some gimmick or another, that doesn't mean they're bad; Just that they require some prediction to work, and if you guess wrong, then the water bunny is going to fall without doing much.
