Most of the refrigerators that you refer to as "built in" are cabinet depth appliances. For an even more customized look these may be purchased with a panel ready option to accept matching cabinet panels for an even more integrated look.

there is not really sucha thing as built-in versus freestanding. except perhaps for the products of the german manufacturer liebherr, whose products can sit INSIDE a cabinet, all fridges stand on their own.
i guess that you are most likely referring to the difference between a so called integrated fridge and a built-in one? of is this more a question that relates to counter depth freestanding fridges versus the regular depth ones?
integrated fridges are meant to be installed in such a way, that the surface of the fridge door is flush with the adjacent cabinetry. That surface can be either the stainless steel finish, or a cabinet door thatcovers the fridge, which mean the fridge ends up being totally integrated, so that nobody can really detect it, other than maybe by the typically larger size handle.
an built-in fridge simply means that the consumer can install a cabinet finish door on the fridge, which when seen dead-on will make the fridge look like the restof the cabinetry. HOWEVER, due to the way the hinges are working on those fridges, the full depth of the door has to stick out past any adjoining gables or cabinets, otherwise you can not open the fridge without hitting and damaging those pieces.

as for counterdepth ( 24") versus standard ( 28-30") depth of the fridge CARCASS ( plus another 3" or so for the door thickness)- those fridges have to stick out just the same way as the built-in one does.
There are a couple of considerations I present to my clients when choosing such a model. We can always order deep enough gables for the fridge to cover the carcass, or pull out adjacent tall cabinetry to hide the carcass, which makes the fridge look built-in. the question if it is worth to go with a counter depth model depends on the clients preference, as there will be a difference in available cubic feet of storage in those models.
It is also a matter of adjacent countertop depths - I differentiate between counters with a typical 1" overhang above the face of the cabinet door, which brings the counter to 25 1/2" depth, and if I want to achieve a clean detail with the counter nicely butting into a fridge gable, then my gable would have to be at least 26" anyways. in this case i will think hard about all my details, and tell the client that they can easily go with a standard model.
when plannng a european style kitchen with a counter without an overhang, which creates a more modern look, it might be worth while to consider a counter depth fridge model.

sorry to be so technical, but good design lies heavily in the details, which, when executed well, are actually NOT obvious...

DO you use a 3/4" end panel or an end panel with 1 1/2" return?