I can't find a clear definition that differentiates sequence assembly vs scaffolding.

From my understanding,

Assembly = joining reads into contigs

Scaffolding = joining contigs into scaffolds (using eg paired-end reads)

Does that sound right? It seems that assembly must be followed by scaffolding, but definitions of assembly don't even talk about scaffolding. Can you do assemble a whole genome with just "assembly"?
Contigs are sequences of overlapping (*contig*ous) reads. Paired-end (or mate-pair) reads can be used to determine the gap between two contigs. When you know the gap, you can make a scaffold, which is just the two contigs with Ns representing the gap in between.

 

EDIT: *Can you do assemble a whole genome with just "assembly"?*

Yes. Scaffolding won't give you more information about the actual bases anyways; it just tries to tell you how your contigs are ordered.
