Do you want true counter depth (~24" deep, flush with counter), or "counter depth" ~27" deep--door is out in front of counter line)?

Both are more expensive than regular refrigerators for the same width, so you are getting less cubic footage for the money. The most expensive will be the truly integrated counterdepth that is built into the cabinetry essentially.
You can easily pay $6000+ for a refrigerator in this category. The least expensive will be a freestanding that is called counterdepth but is really 27-28 inches deep including the handle...who knows why.

I prefer them design wise because I hate to see a hulking refrigerator jutting out and I hate to see the sides of the refrigerator especially when they don't match the front.

However, if you are doing a new kitchen you can recess a conventional refrigerator a bit to get the same appearance.

We have the freestanding 27" counter depth. On the left side of fridge it does jut out beyond the counter by the thickness of the door, but a marked improvement over the old fridge. On the right side of it is a wall that is almost flush with it, so it makes sense for us. Besides it cleared up the walkway a bit from breakfast room to kitchen. Visually it opens up the kitchen quite a bit. For this reason we did not mind paying about $600-700 more. Old fridge is 25 gal, new counter depth one is also 25 gal, but is higher (72" instead of 69"), so we did not lose too much interior space.

To recoup some of the inner volume lost by using a counter depth, look for models that are taller. Some Kitchenaid and GE are taller than standard depth models.

I found my new counter depth Samsung was better laid out than the GE that it replaced. I didn't lose any interior space, but it was about 3" taller. I also didn't pay a huge price for it, bought it on sale for $700.

quite honestly: what kind of PERFORMANCE do you want and are you looking for a refrig/freezer unit combined (be that lower drawer freezer or upper or side by side.
If you are a foodie, it may matter (meaning do you cook ALOT and keeps lots of diff things). Do you use your refrig alot (in and out of it all day). Do you turn over the contents quickly or do you store more bottled, prepared things than fresh?
Certain higher performing units only come counter depth or near counter depth. You pay both for the privilege of not having a hulking box but, yes, better performance... meaning, lower noise, ease of access/well designed, holds its temperature well in the various zones etc. I have an Amana at parents place in Calif and it is Just OK (tho i love the pull out freezer drawer)tho hulkingly large; we have a kitchen aid regular similar hulk in a cottage and this is refrig #2 --- very disappointing that it already, after 6 yrs., makes noise --- and we have a Subzero purchased in 1993 that continues to be truly wonderful...a joy to use, to clean. No noise and keeps food super fresh.

I have the counter depth where the doors are the only things visible at the side. It was crucial to the floorplan, or it would have interfered with a door. The space inside is curtailed compared to a regular depth model, so if there is any way to recess yours into a wall pocket, that would be the best outcome; you get flush appearance and full-size cu.ft.
Casey

Thanks everyone.
I guess I didnt realize there were two options available. I know the $ will be more but for the design and room My KD thought it would be best. I am thinking of french door style is the kind Im interested in.

I've been looking for a French Door Refrigerator that is cabinet depth, built in flush, fully integrated refrigerator. I'm hearing that to date, there are none. They are all 25-26 inch depth, so they stick out past your cabinet.

Kimmine, not sure what width you are looking for, but Jenn Air is coming out with the first French Door flush built in...(a 42" width model) supposedly this September, but they aren't promising and I'm running out of time.

Sombreiul, which refrigerator do you have? I have one more month to make a decision and I'm probably going to have to go with a built in flush side by side.

Here is a link that might be useful: First Flush French Door Built in Refrigerator: JennAir

I know I'm in the minority but I wouldn't buy a CD model again. We bought one about 5 years ago and sold it a year later. The lack of space drove me batty and I never liked the unbalanced feel of the heavy door hanging on the shallow cabinet. Funnily enough I never minded giving up the extra floor space when I replaced it because those few inches were barely noticable, even though our refrigerator faces a fairly narrow path through our kitchen. Looking back it seems illogical I paid a premium for less capacity. I was glad there was still plenty of demand when I sold my CD though because the resale value was very good. Live and learn.