Ability drain and ability damage are both forms of ability score loss, but they are separate types of that. Immunity to one does not confer immunity to the other.
And backing that up is hard other than to point to the distinct lack of any reason to think it would. Nothing says they are the same, they have different names, when the rules want to refer to the two of them together they use yet another name (ability score loss) to cover them as a pair, and different effects cause immunity to one or the other or both of them.
For example, the undead type has
- Not subject to [...] ability drain [...] Immune to damage to its physical ability scores (Strength, Dexterity, and Constitution),
If ability drain and ability damage were the same thing, the immunity to physical ability damage would be completely redundant, and the implied vulnerability to mental ability damage would be false because those, too, would be covered by the creature not being subject to ability drain.
When the authors wanted to define things as being different types of the same thing, or one thing being a specialized form of another thing, they knew how to write that clearly. For example, ability burn’s description starts with “This is a special form of ability damage,” while ability drain and ability damage say nothing of the sort about one another.