One difference is that inset cabinets are more expensive. Here's something that helped me understand:

Here is a link that might be useful: Cabinets

They are two attributes that are not mutually exclusive, as "framed" pertains to the carcase construction, and inset pertains to the way the door/drawer is mounted to the box.
The alternative to framed is frameless; the alternative to inset is overlay, be it partial overlay or full overlay. The amount of overlay is sometimes expressed in inches, like 3/4" overlay or 1 1/4" overlay, or as a fractional ratio; half overlay or full overlay. Overlay is often coupled with framed cabinet boxes, but frameless cabinets can also be made with inset doors.
Inset refers to the doors and drawers being diposed so that either the full faceframe or the edge of the box (frameless) is visible when they are assembled.
Two varieties of inset cabinetry are typically encountered; where the doors fit into a plain undorned frame, or where they fit into an opening that is decorated by a bead molding; the latter is called "beaded inset".
HTH.
Casey

We are getting inset. The biggest drawback is the additional loss of space, but I'm willing to do it for the look I love. As for being more expensive, if you decide to go with inset, make sure to shop around. We got several bids, for two of them it would have been an additional $7000. For another it was $2000 more, and with the cabinet maker we are going with we are paying only $1000 extra (and we have a fair number of cabinets). I'd shop around in any case, but especially if you want inset.

I hope that helped. Inset cabinets mean the doors and drawers are 'flush' with the front face framing. The doors don't overlap the opening like an overlay door, instead, they fit inside the surrounding border - the front 'frame'.

European cabinets, or "frameless" have no front 'face' or front frame border to the opening, just the edges of the sides. A door, or drawers make up the entire front, usually covering the narrow side ( and top and bottom) edges.