To add a little to the valuable replies already given...
The first feature that is very characteristic of centrifugal compressors, and which is - to my opinion, a game changer relates to the so-called flow coefficient. Because generally speaking, gases are compressible, the actual volume of gas reduces through the successive compression steps. So the flow passage (impeller exit section) becomes narrower from one impeller to the next as the flow proceeds though the machine. This is something that can be observed physically; for instance a H2 compressor which operates upon a gas that is typically "hard" to compress will have successive impellers of exist sections that are almost equal while the differences in impeller's width is immediately noticeable on, say a Propylene compressor or even a natural gas compressor.
Ultimately, when suction inlet flow shifts when compared to the design flow, not only it offsets the 1st stage operation with regard to the best efficiency point and available margins to surge/choke, but that very effect stacks up across the successive stages and becomes quite dramatic on the last stages. All in all, the entire machine performance (efficiency, turndown) can be heavily effected by any shift occurring at the inlet. The phenomenon can be more or less accentuated depending upon the shape of the curves under consideration and the available turn down. As example, a CO2 machine would typically exhibit quite steep curves with narrow turn down / operating range. A similar outcome can happen due to fouling (take as example a coker gas compressor); because fouling narrows the flow passage section, even though the operating conditions remain at design, the machine / operating envelop is not anymore fitted to the conditions it has been designed for; performance degradation is one notable consequence. Some of the common implications encountered in contract negotiations have to do with guaranteeing performance outside of best efficiency area because of the aerodynamic peculiarities above described.
The second feature that is also remarkably specific to compressors and which you would not encounter in pumps, is that the compressor performance curves are set for a specified set of inlet conditions (namely pressure, temperature and molecular weight). Thus when a machine operates at different inlet conditions than design, a new set of interpolated curves is required based on the inlet conditions under consideration. Some techniques for estimate do exist, but in general you are left to the diligence of the OEM to provide you with the needed input, when not available in the original set of curves.