Q: Why is it, "It seems different", but "It comes across differently"? Both phrases describe the manner, appearance, air, etc, of a subject. Why does the former use an adjective to modify the subject, while the latter uses an adverb to modify the verb phrase?

A: That is purely a matter of syntax, not of meaning. The verb to seem happens to be a copula, which normally has an adjective or noun as a secondary complement.

  
*
  
*You seem sad.
  
*She seemed a good doctor. [This might not sound as natural, but it is
  still acceptable.]

The verb to come (across) is not a copula; that means it does not normally have an adjective or a noun as a complement or adjunct, but instead an adverb, as most verbs do. (In informal speech, you might occasionally see come across used with an adjective, because some people intuitively treat it as a copula.)

  
*
  
*You come across differently.
  

However, if you use come across with as, it gets an adjective or noun.

  
*
  
*He came across as a bit stubborn.
  
*She came across as a bully.
  

If you use seem with like, only a noun will do.

  
*
  
*She seemed like an easy catch.
  


A: It's not really exact to say that both sentences describe the manner, appearance, air, etc...
It's better to say that one modifies the object/subject, the other "describes" the manner, how the action is being performed.
In the first example, "different" modifies the subject, what is indicated with "it", but in the second one, "indifferently" modifies the verb, not the object/subject. 
This is the usual way those two classes work, adjectives and adverbs, although there might be exceptions. I can't come up with any at the moment.
