We use these two words as if they were synonyms.  However, originally, the signature was the tiny letter (in UK) or number (in USA) which was printed at the bottom of the outer page of the folded “signature”.  The folded piece took on the name of that little mark.  In the world of sheet-fed printing one signature might well be inserted into the middle of another signature to create the section which would form the basic unit of construction of the book block.  Thus the bindery would tend to talk sections, while in the pressroom and folding department it was more likely to be signatures.

Each section will usually have a collating mark, commonly a black bar, printed on the spine fold of the outer pair of pages in the section.  These collating marks are placed so that when you look at the spine of a gathered book, they step down in a smooth diagonal from top to bottom.  This provides warning to bindery workers if something is out of sequence.  If it is, there will be a jump in the smooth line of marks moving down the spine from top front to bottom back.  As the practice of inserting signatures to make full sections fell out of use, those little signature marks migrated from the bottom of the first page of the sig and became incorporated into the collating marks on the spine.