How are arms different than legs?
I think that's more of a matter of linguistics rather than any scientific/physiological reason. For instance, in Portuguese, you wouldn't call legs to a cat or a dog's limbs. You would say they have 4 "patas" (roughly translated as "paws"). And calling them legs, in Portuguese at least, would be weird.

When veterinary doctors refer to animal limbs, they usually use anterior and posterior limbs. In fact, anatomically speaking, anterior limbs in cats and dogs (and probably in other animals too) have more in common with human arms (upper limbs) that with their own posterior limbs.



Also, as rg255 commented, maybe the reason behind the colloquial nomenclature is quite simple... 

To (miss)quote Nancy Sinatra: 


  These "legs" are made for walking
  
  And that's just what they'll do