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
- How about the red one?/What do you think about the red one?
can be understood (from context) as
- what is your opinion on the "red one", or
- let's choose the red one
- "How about" is a phrase and is more informal. It is also more suggestive (but that still depends on emphasis and context).
- "What do you think" can be taken literally e.g. What do you think about the game last night? as inquiry about opinion, maybe as start to a longer conversation; in the same example the other phrase works very well, too: How about that game last night(!). However, notice the change of punctuation - typically you would use "how about" if the game was exceptional in some way.
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.
I would say 'How about' is more usually used as a follow up question, an exchange may be: "What do you think of the white one?" "I don't like it." "How about the blue one?" "Oh, that's much better"
However, in less formal usage using 'how about' as short hand for 'what do you think of' seems to be becoming more common.