What's the difference between these two? Is мама considered "childish"?
Not at all. "Мать" is "mother", "мама" is "mom", and "мамочка" is 'mommy' respectively, with word preference following the same rules (formal vs. informal) as in English. Usually, the way one refers to their own parents when talking about them ("мама"/"папа" vs. "мать"/"отец") quite clearly indicates the actual closeness of their relationship and the level of comfort within the family. Less intelligent folks may even go as low as "мамка"/"батя", but those are almost derogatory. Children always use мама/папа no matter what.
In official context or when talking about non-human subjects, 'мать' is always used.