I have to assume you asking this with respect to a refrigeration condenser or evaporator application. I also assume that the tube coils would be radially finned on the air side (outside the tube). The main features of such a heat transfer unit in each of the two configurations are:
Counter flow:
1) A shell (or shroud) is required to ensure that the air is routed across the tubes; this requires some sort of multiple baffles - and this translates to a relatively large pressure drop or high pressure requirement for the air.
2) This configuration is by far the most efficient and requires the least amount of coil length and materials; it also requires the least amount of air for the condenser;
3) With the high efficiency however, comes an increased difficulty in carrying out external tube (or fin) cleaning and inspection; the necessary shell or shroud are the culprits here.
Cross-flow:
1) This configuration is probably the least efficient; you only get one shot at passing the air across the tube bank;
2) However, it is the easiest to fabricate and assemble - as well as to inspect and maintain.
3) Another advantage is the relatively low pressure drop across the tube bank; this makes it very appropriate for the evaporator application where you only have inches of W.C. in pressure to give up.
4) The configuration has an inherent advantage in being able to be insertable in the common air ducts - to act as an evaporator - with a minimal of sheetmetal work and modification required.
I can come up with some more but the above, I think, are the major differences in characteristics.
