Melamine is the (usually) white covering used on MDF. IMO, it's inferior to plywood box (not sure what "furniture board" is). However, the frameless construction gives you bigger openings for the drawers and doors (nothing to get caught on other than the counter edge) while the framed face will have things to catch everywhere.

Furniture board is high grade composite wood used in cabinetry. It’s more moisture resistant than either standard particle board or plywood. It is the ONLY thing used to make the high end European cabinets, and has always been accepted there. Here, uninformed wood snobs look down on the better performing composite products. Usually because someone trying to charge you more for an “upgrade” has sold them on that snobbery.

Plywood will drlaminate quicker than furniture board will disintegrate. But it generally hides the damage better, all the while the structural strength is gone.

Plywood vs Furniture Board · More Info

There are good better and crappy grades of furniture board, plywood and MDF. China helps with that a lot. If you can find frameless with Blum Tandem Blumotion undermount full extension drawer glides with soft close in quality ply or furniture board construction, . . .Frameless boxes need to be solidly built because they don't rely on a face frame for structural integrity. Ikea makes boxes at a fair price. Frameless only. You can use their drawers/doors or get them from Barker Here. If you have Amish cabinet makers anywhere in your area that can be a good value choice. Barker's frameless boxes are US made plywood RTA with Confirmat Screw Assembly. More info.