Does the tooth spacing of the cutting blades make a difference?

Yes. The more space there is between the teeth, the larger the branches or twigs that can be cut. 

 

As you can see from the diagram below, the spaces allow branches between the blades so they can be cut more cleanly. 

 

 Needless to say, the more powerful the machine, the more it will cut through, however hedge trimmers are not chainsaws and are suitable for cutting branches of up to 25mm or so.

 

A good guideline is –  if the branch is much thicker than your thumb… don’t try to use hedge trimmers, they will probably stall or jam.