Both are types of conformity, which is when we change our behaviour to fit in with the majority group. The main distinction between compliance and internalisation is whether or not our personal belief changes too.
Compliance
We conform to the majority to avoid being disliked by others.
We conform to gain their approval but our private belief remains the same.
E.g. Your flatmates tell you they’ve turned vegetarian and list several reasons why. You don’t think they are particularly good reasons but you don’t want to them to dislike you so you stop eating meat too.
Internalisation
We conform to the majority to avoid being incorrect.
We observe the behaviour of others and reassess our own beliefs. We decide that their belief is correct and alter both our behaviour and beliefs.
E.g. Your flatmates have turned vegetarian and explain the reasons why. You decide that they have a valid point and that being vegetarian is better than eating meat so you also become vegetarian.