Generally, the British call all suspenders braces, but in the United States, there is a difference between the two based on the type of closure used to attach them to a pair of trousers.
In America, suspenders usually refer to the fabric or leather straps with a drop clip end that attaches to a pair of trousers. Sometimes in British English, however, suspenders refer to what Americans call garter belts, which are used to hold up stockings.
According to some American companies, braces usually refer to the leather or cloth strips with button ends that attach to trousers. For the British, however, braces refer to all suspenders.
Usually there is no distinction between braces and suspenders except for a few select retailers. Most companies just recognise them as ways that two different countries refer to cloth or leather strips that are worn over your shoulders and hold up your trousers.