Reciprocating compressors are very efficient, very rugged, and very well understood by field operating people. Single stage recips are the baseline for all compression adiabatic efficiency calculations. I use 4.5 compression ratios per stage as the maximum design conditions, although specific machines are designed for somewhat more or less. Two stage machines require about 6% more Hp per cubic foot compressed. Three stage machines need about 14% more.
A recip will develop very high temperatures on the cylinder outlets that must be accounted for in the skid design. The weakest component in a recip is the valves - compressor manufacturers like for suction pressure to be within +/- 5% (in absolute terms) of the design conditions which is doable in plants and pretty tough in field non-air service (in most compressed-air applications the suction pressure never varies much at all).
What I see as the other main drawback to recips is balancing stages. If you get a machine set up (i.e. the right valves, clearances correct, etc) for one set of conditions and then suction or discharge changes you often end up doing all the work in one stage and really reduce the capacity of the machine while increasing the risk of mechanical failure. I like to use 40 psig as the lowest suction pressure where a recip is my first choice.
The two most common rotary compressors are centrifugal and screw.
Centrifugal compressors are very common in service where the discharge must be oil-free (recip's inject oil into the process stream to lubricate the cylinders). It is an easy thing (during design) to stack stages to get any desired ratio of skid suction to skid discharge pressure. Temperatures must be carefully managed across the stages. A centrifugal is about as efficient as a 3-stage recip. They are often used at the inlet to a process where oil contamination would be a problem. These machines are also used extensively in places where energy density is especially important (e.g., since a centrifugal can be driven by a gas turbine the hp/ft2 of deck space is very high and they are often used on offshore platforms).
Screws come in dry and oil-flooded.
Dry screws are slightly less efficient than a centrifugal and are typically used in small applications where a centrifugal would be too large. Dry screws connect the male rotor to a driver and use a timing gear or chain to drive the female rotor. They have all the temperature problems of a centrifugal, but the high temp is limited to the single discharge stream. A dry screw can do up to about 6 compression ratios. They can more easily adapt to changing operating conditions than a centrifugal since the screw is positive displacement and a centrifugal is not.
Flooded screws have been around for about 30 years and are my first choice for suction pressures below 40 psig. They have an oil flood that seals around the rotors, lubricates surfaces, acts as a cushion that allows the male rotor to drive the female without a timing gear, and carries heat away from the screw. Ten compression ratios will increase oil-temperature about 35F in a flooded screw while the temperature increase in a dry machine for 10 ratios is closer to 380F. I use flooded screws anywhere the suction pressure needs to be between a moderate vacuum (say 12 inHg) and 40 psig and suction conditions could reasonably be expected to fluctuate. They can usually handle 10-15 compression ratios without much problem.
Flooded screws are just slightly less efficient than single stage recips (the oil seal and the lower temperatures account for the dramatic improvement in efficiency). The biggest drawback is is the oil. It is expensive, you have to carefully manage temperatures to keep from collecting water in the oil, separation of oil and gas is harder than most skid packagers understand, it takes a non-trivial amount of hp to move the oil, and it is really a pain to deal with if it gets off the skid.