It seems that both Ansicht and Meinung mean opinion. So what’s their difference?
It's the same difference as point of view (Ansicht) and opinion (Meinung).
eg. Meiner Ansicht nach translates to in my (point of) view but Meiner Meinung nach translates to in my opinion. But both is often used interchangeably both in English and in German.
Let me quote from a blog article I found about this:
What you see or think at any moment comes from a point of view. How you understand it in relationship to other objects or ideas is your perspective. What you state about it beyond simple fact is your opinion.
I wouldn't be able to describe it any better and the more I think about it, the more it seems philosophical.
Jaschas answer is correct but misses the fact that Ansicht may also mean a view in general, i.e. if it is not used as in point of view. An example would be Ansichtskarte, a postcard with the view of a panorama on it.