Belts and braces do the same mechanical job, which is keeping your trousers where you left them, and they are two quite different choices in how a suit ends up looking. A belt is practical, adjusts in a second, and is the right answer for most men in most situations. Braces, worn right, are more comfortable and more formally correct, particularly with a high-waisted trouser, with a formal suit, and for a man who has reached the point of knowing exactly what he likes. We keep both here to go with the trousers we make.
The honest case for each
A belt runs through the loops and buckles at the front. It is practical, instantly adjustable, and perfectly correct for almost anything casual and most things professional. The argument against it at the dressiest end is simply that it draws a horizontal line across your waist that cuts the line of the suit in half, and that a trouser built for braces, with buttons instead of loops, hangs cleaner without it. That is a real point, not a snobbish one, and you can see it in a mirror.
Braces, or suspenders if you prefer the American word, button or clip to the inside of the waistband and hold the trouser from your shoulders rather than gripping your waist. The trouser hangs from above, so it falls in a straight line from waistband to break with no gathering or bunching around the middle when you move. For a high-waisted or Gurkha trouser they are the right choice. With a formal dinner trouser they are the traditional one. And under the waistcoat of a three-piece suit they are what has always been worn, for the good reason that nothing else keeps the line as clean.