The meanings are the same. The intention of each is slightly different:
In a) the sentence is describing how fat the person was. The intention is to inform the listener of the extent of the person's "fatness".
In b) the sentence is describing an outcome as a result of the person's "fatness".
If they were answers to questions, the questions would be
A) How fat was he?
B) Why couldn't he get through the door?