Yes, Constance. Person B is happier than person A.
The construction ''as + adjective/adverb + as'' is used to make comparisons when the things we are comparing are equal in some way. For example: The weather this summer is as bad as last year.
On the other hand, not as … as is used to make comparisons between things which aren’t equal. For example: His new flat isn't as nice as his old one. It means that his old flat was nicer than the new one.
I hope this helps :)