Is there a terminological difference between "sequence" and "complex" in homology theory Suppose you are given something like this:
$\dots \longrightarrow A^n \longrightarrow A^{n+1} \longrightarrow \dots$
People tend to talk about "chain complexes" but about "short exact sequences". Is there any terminological difference or any convention with regards to using these words (EDIT: I mean "complex" and "sequence" in a homological context) that a mathematical writer should comply to?

To say that the sequence is a chain complex is a less imposing condition: it simply says that if you compose any two of the maps in the sequence, you get $0$. But to say the sequence is exact says more: it says this is precisely (or, if you rather, exactly) the only way you get something mapping to zero. 
The first statement says that the image of the arrow to the left of $A^n$ is contained in the kernel of the arrow to its right, the second statement says that the reverse inclusion also holds.
