So, what is the difference between Braces and Suspenders (if there is a difference)

Braces

If you call suspenders braces, you are probably British, or into tailored clothing. Braces is the term the Brits use for the accessory and traditionally the term refers to Button-on suspenders. These button-on suspenders are generally made of higher quality materials and intended for business and dress wear. 

Above: Button-On suspenders (or braces!)

Suspenders

Suspenders, on the other hand, can be fastened with metal clips. For some, this is the distinction between the two. This is mostly an American term as the Brits use it to describe what we’d call in the US, a garter belt to hold up hosiery.

Clip suspenders

Above: Clip suspenders

Our conclusion: 

As you can see there is not really a clear big difference between Braces and Suspenders. We do not care what you call them, as long as they do the job, holding up your pants!