The all-mediums configuration has the advantage (so it's said) that you are
more often shifting entire boxes than individual frames.  That is one
reason why Dr. C. L. Farrar used and recommended them decades ago.  His,
incidentally, were more or less square, and held 13 frames of Illinois
(medium) depth.  He overwintered colonies in stacks of 4 or 5 of these.
 
I recently came across a new German beekeeping manual in which the author
devoted an entire section of the book to the management scheme using
mediums only.  It was the same basic idea -- you shift around entire boxes
to direct the activities of the colony. Evidently this is catching on in
Europe.
 
The smaller units are a great deal easier to move around.  The drawbacks
are, you have 30 or 40 brood combs, so it can be a royal pain to find the
queen.  The investment is more, as you need a higher number of boxes and
frames to achieve the same volume as with deeps. This can get expensive, as
frames and boxes are about the same price whether you buy deep or medium
size.
 
Burr combing, or connecting of the combs in one box to the ones below and
above, is a bane here, too.  The bees (evidently) prefer continuity in
their brood nest, so the spaces between brood boxes get filled with drone
comb and brood, even though the bee-space is "correct."  This happens with
deeps, too, of course, but perhaps moreso with shallow frames and numerous
shallow brood-boxes comprising the brood chamber. There is more wood within
the brood nest, too, and some have claimed this restricts the queen's
rounds, and as a result the compactness of the brood pattern. Please relate
any experience along these lines.
Thanks... JWG