Why do turbine engines take so long to spool up?
I'm not an engineer (which might be better suited to answer this) so this is from simplified things they feed pilots:

Jet engines take much longer to spool up (i.e. increase RPM) than piston engines, especially at low RPM because of the pressure ratio/increased airflow necessary to keep the compressor from stalling/surging/blowing up every time when changing power settings.

The (simplified) jet engine cycle contains a compressor that pushes air into a combustion chamber, where it burns, and then blows out the rear end rotating a turbine that moves the compressor from back where we started at. 

If you "add more power" (i.e. put more fuel in) it takes a while for that extra fuel to produce more thrust, which in turn takes a while to accelerate the turbine, which will make the compressor spin faster, which will finally bring more compressed air into the combustion chamber to utilize all that extra fuel you poured in at step 1. 

Adding power all of a sudden will increase the pressure in the combustion chamber, so much that air that is "upstream" (i.e still in the compressor) does not want to advance. The extra pressure in the combustion chamber did not have enough time to spool up the turbine, so now the compressor is underpowered to keep "pushing" compressed air into the combustion chamber. Air starts flowing backwards (i.e. from combustion chamber to compressor), engine surges, all hell breaks loose.

So there is a lag (electronically these days, the pilots can floor the trust levers as fast as they like) when the engines are at low RPM, the FADEC only adds a little bit of extra fuel, waits for the airflow to stabilize, then adds a little more, and so on.

I think the graph below might explain this. Every time you change RPM, you will increase Pressure Ratio (i.e. move up the graph), then you wait a little bit for Air Mass Flow to increase (i.e. move right). If you increase the Pressure Ratio too much, without the accompanying mass flow (which take a while because of inertia) you will enter the surge line.



A similar thing happens on spool down, albeit a gentler one.

Compare this with the piston engine where you put more air/gas mixture into the cylinder, it makes for a bigger bang, accelerates the piston faster, and the very next piston cycle you can theoretically get max power.