One of the possible reasonsI know is that capacitors are available in wide range of values and can be made more accurate than inductors. What other factor leads to the use of capacitors over inductors?
A lot of circuits require medium/high impedance values for even moderately low frequencies such as audio and a capacitor of say 10 nF at 1 kHz has an impedance of 15.9 kOhm. An inductor having this impedance at 1 kHz would have a value of 2.53 henries.
Now that isn't a small value to fit in a space for a surface mount component. That's one reason and the next is cost - try finding a 2.5 henry coil in farnell, digikey or mouser and see how much change you have out of a dollar or a pound. The 10nF will cost you about a couple of pennies maximum.
So, it's not small and it's not cheap and a side effect of it not being small is that it will possess significant parasitic capacitance (several pico farads if not more) and this makes it less than perfect. I'm certainly not saying caps are perfect but they are a couple of orders of magnitude more perfect than an inductor.
Also it will have a significant DC resistance as it is made as small as it can be. It won't be good at handling currents like capacitor is and the core will probably saturate.
Bad idea - use capacitors.
Generally speaking inductors are much more lossy than capacitors. They depart much more from the ideal models that people learn at college, and in a poorly specified fashion. In other words a circuit having inductors instead of caps is more likely to need tweaking.
Also inductors, depending on how they are constructed, are more prone to picking up stray fields, which means more PCB layout constraints. The whole process from hand sketched circuit diagram to working PCB is more predictable when inductors are not there.
Finally, if you need to substitute a part, you are more likely to have problems with an inductor than a cap, as many inductors use non-standard footprints.
Both coils (inductors) and capacitors differentiate or integrate something w.r.t. time.
I think, in almost any context, an active circuit (one with op-amps or some kind of transistor amplifier) with inductors will cost more than one with the equivalent number of capacitors. It takes more to manufacture a coil than it takes to manufacture a capacitor.
It would be possible to build integrator or differentiator circuits using inductors, but these would integrate current instead of voltage.
You would need a current amplifier (low input impedance, high output impedance) instead of a voltage amplifier in this case as well. These are somewhat more impractical to build.