A website template/theme controls the website branding - the "look and feel". Different platforms refer to one or the other, but they mean the same thing.
A template/theme controls website elements such as layout, logo and navigation positioning, mobile layout, fonts, colours and other styling schemes.
When we start a project, one of the first questions we'll ask is for you to provide your current branding, as well as some existing websites you like, perhaps competitors or just a favourite business. This helps us get a feel for the type of "look and feel" you envisage.
What To Look For
The most difficult thing for most people when looking at template/theme demo, is to try to visualise it with their own words, colours, photos and logos - it's really hard!
The rule of thumb is to try to find a template/theme which is mostly similar to how you'd like your own site to look and function. While most things can be changed, the more changes that are required, the longer it takes and the more it costs.