These two sentences seem to be the same to me. Is there any difference between them, and are there circumstances in which I can only use one of them instead of the other?
Both of them can be used perfectly in cases when you want to present a choice or if you actually want to ask someone's opinion on something.
For example, both versions
can be understood (from context) as
A solid answer to this is more complex than it might at first appear.
How about/What do you think about (Michael?).
When the phrase is followed by a single object - could be a person or an item or an action in the gerund - you are offering a suggestion from given options. In my example, perhaps Michael is one of several applicants for a job.
How about/What do you think about (Michael telling Sarah he loves her?)
Here, the phrase is followed by a full clause. In this case, while it could be a suggestion depending on context (Should Michael leave Sarah or tell her he loves her?), it is more likely that you are asking for opinion. Used like this, it often finds itself in gossip based conversations, as you can see in my example.
What do you think of (Michael?)
Since you didn't include the preposition in your question, I'll add this. Using 'of' instead of 'about' only works with 'what do you think...' and not 'how...'. It is used to ask of your opinion about the object. Tell me your opinion of Michael; do you like him?
Hope that helps.