I've had experience with both floor types and each offer benefits.
You wouldn't go wrong with either.
Polished concrete is a little bit of a misnomer, since while concrete
can be finished to a sheen (called hard troweled), a true gloss or
polished sheen is best done by having a sealer coat applied. There are
many 'sealers' available and you will get what you pay for. Two-part
seal coat finishes are best, but usually cost more and are a little more work to prepare prior to application but are very doable. The two-part sealers are available as water-based mixes which is important.
Color can be attained by different methods. A few choices are 1)integral color (color is part of the concrete mix), 2)acid etched into a concrete surface (color is a result of an acid based material sprayed onto an alkaline surface-in this case concrete), or 3)surfaced applied (usually sprayed). With any of these coloring methods it is be best to have a sealer applied as the last step. The sealers are available in flat or matte, semi-gloss, or gloss or "polished" sheens.
Wood floor types vary. I'm most familiar with 3/4" oak, tongue &
groove. 1/4" top nailed is very common in older homes.
The sealing issues with wood floor are similar to concrete floor;
finishes are available in various 'sheens', and one-part or two-part mixes. Colors available are numerous and are surface applied prior to sealing
similar to what is done with finishing furniture or other wood material. Clear sealers allow the beauty of the wood to be seen, while clear sealers on plain or non-colored concrete might not be what you want.
Durability is as different as the difference with concrete or wood so
concrete will last forever. The type of sealer applied on the
surface really affects the durability of either material (especially wood), and wood is more susceptible to excess surface moisture. The substrate under either material is important and can ultimately affect the longevity of either. If you are going with concrete make sure substrate/subfloor can handle it
Re-finishing of either wood or concrete involves a fair amount of work but is possible with each. There is a limit to the number of times a wood floor can be refinished.
Cost for either material can vary widely and is dependent on a number of things; type of wood, thickness of concrete, room layout, condition of substrate, when completed (new or rehab construction), etc. Wood is probably lower cost or at least can be more competitive since there are many more installers (can also mean more lower quality installers).
Lastly, there is a certain 'warmness' to the appearance of a wood floor that some think is not possible with a concrete floor, though radiant floor heat (either electrical or hydronic) is possible with either.
Again, I don't think you can go wrong with either material. When installed in a professional manner and proper prep conditions, either concrete or wood can give you years of enjoyment. Best of luck with your decision.