Azurill will have more attack than Lillipup. Huge Power doubles their actual attack stat, not their base attack stat.
At Level 10, if both have 15 IVs, no EVs and a neutral nature, Azurill has 20 attack, while Lilipup has 18.
Here's the stat comparison actually:
Lilipup: 30 HP, 18 Atk, 15 Def, 11 Spatk, 15 SpDef, 17 Speed
Azurill: 31 HP, 20 Atk, 14 Def, 10 Spatk, 14 SpDef, 10 Speed
Azurill's only problem is that it's slow (which is really its main problem throughout the entire game).