What is the difference between using an USB Logging Software and Logic Analyzer for signal sent to the printer?
A logic analyzer just grabs the raw digital ones and zeroes but can do it for almost anything. You need to do the decoding work yourself, more or less.

A protocol analyzer can only grab data for one protocol, but will do a much more extensive job of it and help you interpret it.

Some logic analyzers do some decoding for you, but it's not as good or as extensive. You also probably have to set it up in the logic analyzer software yourself manually.