In sentential logic, the smallest things you evaluate are whole sentences. In first-order logic, you can break up whole sentences and look at their parts. To see the difference that this makes, consider the argument:
All humans are mortal.
Socrates is human.
Therefore, Socrates is mortal.
This argument makes sense (I hope!). Now, if you only look at whole sentences, you can replace each sentence with, say, a letter. Let's replace All humans are mortal with A, Socrates is human with H, and Socrates is mortal with M. The argument is then:
A
H
Therefore, M
The argument has lost the information that made it make sense. We need to look inside of the sentences to see why the argument makes sense. If we just replace the parts Human with H, Mortal with M, and Socrates with S, the argument becomes:
All H are M.
S is H.
Therefore, S is M.
Now we can see how the parts of the sentences make the argument work, and that's the major difference between sentential and first-order logic.
BTW, I simplified:
All H are M.
S is H.
Therefore, S is M.
It would look a little different in first-order logic, but explaining the details would've just gotten in the way. For instance, All Humans are Mortal would be stated \forall x (Hx \rightarrow Mx). The idea is still the same.
I presume deductive means syntactic. From a syntactic "point of view", you are looking at what you can prove, i.e. whether you can use some set of rules to derive a sentence. You may use the rules to derive a sentence from another sentence, several sentences, or even from nothing. So syntax is associated with proof and proving.
From a semantic "point of view", you are looking at whether a sentence is true or false. You may consider whether a sentence is true or false by itself or given that some other sentences are true (or false). So semantics is associated with truth and deciding.
The metatheorems could be several things, and as you can imagine, are mostly too involved for a quick explanation.
I don't know how much that helps. If something wasn't clear, just ask, and I'll give it a go. :)
