The answer to this question is based on liking or preference, which varies depending on one's affinity to a particular taste.
The texture of both these vegetables is quite different. While the former is a member of the corn family and is a versatile vegetable which is extremely low in carbohydrates, the latter is a savage rustic vegetable, quite low in fats.
Mushrooms are of various kinds, some are brown, others are white. But whichever type the mushrooms be, they are very plain and bland in taste.
The same goes for Baby corn, I feel. These vegetables are tasteless unless they are either frozen (which imparts them a sweeter flavor) or taste as per the spices which have been added to them while cooking.
Some people like mushroom due to its water retaining capacity, low-fat content and mild softness, others prefer baby corn due to its even softer texture and a rich flavor.
Thus, choosing one out of these two is a matter of one's decision to have what kind of flavor tantalizes their taste buds that very day.