i think they are basically interchangeable. Looked up some dictionaries, they did not indicate that 毛病is informal at all. It is not in my experience either. Further, i don't think there're anything wrong with saying 他的理论／思想有毛病。 I will translate 毛病 into defects and 问题 into problems and I feel personally i tend to avoid saying someone has 毛病，it sounds a little bit arrogant while 问题 is softer. Is it the same with the english expressions defects and problems? I'm not sure but I'd feel more comfortable to say to my boyfriend "you have a problem" rather than "you have a defect". could be misunderstanding for sure. ah, you can also use 问题 in this way: 健康问题；生机问题. 问题here means issues. "personally, i won't say 健康有问题either" 健康just don't sound it collocate with 问题' but I could be wrong!

Agree with Zander. A bit more to add.... As 毛病 is an informal expression, you cannot use 毛病 with a formal word. For instance, his theory is problematic. People say 他的理论有问题, but not 他的理论有毛病。 When saying health, whether 毛病 can be used depends on particular topics. People say 他身体有毛病，but will not say 他的健康有毛病。他的健康有问题 is more common.

The problem with this question is ... 这个问题的毛病在于。。。

毛病比较口语化，不能用在正式场合。其他场合基本来说‘问题’和‘毛病’可以互换。Besides meaning problem, they also mean drawback, shortcoming(缺点) 例如，这个车最大的缺点\问题\毛病就是费油。 毛病 also mean bad habits(坏习惯) 例如，这个孩子最大的毛病就是吃完糖不刷牙. :) Hope it will help