Belt vs braces — the practical and formal arguments
The belt goes through the belt loops of the trouser and buckles at the front. It is practical, immediately adjustable, and the correct choice for almost all casual and most professional settings. The formal argument against the belt in the most elevated dress is that it adds a horizontal line at the waist that interrupts the line of the suit or trousers, and that a trouser made for braces — with button attachments rather than belt loops — hangs more cleanly.
Braces — suspenders in American English — attach to the inside of the waistband via buttons or clips and hold the trouser from the shoulder rather than cinching the waist. The trouser hangs from above, which allows it to fall in a clean vertical line from waistband to break without gathering or bunching at the waist during movement. For a high-waisted or Gurkha-style trouser, braces are the correct choice. For evening dress with a formal dinner trouser, braces are traditional. For a classic three-piece suit, braces under the waistcoat are the historically correct choice.