Hello there,
Well do be fair, if you use a power supply of plus and minus 10 volts for example then you have to use a single supply of 20 volts. That means the op amp gets almost the same bias anyway.
But i dont think you'll see too much difference, except in some areas which i'll mention here.
First, there is usually a limit on the output voltage as to the max and the min that it can put out. This means the upper range can be severely limited on a plus and minus supply while it will not seem to be as much with a single supply. For example the LM358 is spec'd to output up to Vcc-1.5v, which means for a dual 10v supply with the output ranging from 0 to +10 we'll only get 0 to 8.5v, and that's 85 percent of the positive rail. But with a single supply, we'll get about 0.1 to 18.5, and that's 92.5 percent of the positive rail.
Second, the same goes for the lower end, but is less severe in some cases. For an op amp that can put out 0.1v up from ground, that means the dual 10v setup can actually put out 0v while the 20v setup can only output 0.1v. This actually becomes a big problem in some single supply setups.
Also, some op amps cant even get that close to either rail, so there the output range is limited even more so the choice of supply could make a difference whether the circuit works at all or not as intended.
We can also say roughly the same thing about the inputs. The input ranges is usually limited too so we have to choose carefully sometimes.
If you are interested in the actual running characteristics then set up an op amp of your choice in both circuits to act as an amplifier. See if the bandwidth changes at all for either power supply choice. The internal circuits are usually biased with constant current circuits so there probably wont be too much difference, except maybe for power op amps. Do some measurements and compare.
ADDED LATER:
Oh yeah one more thing, since the output slew rate is more or less constant that means an op amp that has to slew 20 volts will take twice as much time to reach max (or min) as that which only has to slew 10 volts (to zero or max). Of course if the dual 10v supply op amp circuit has to slew from max minus to max plus supply then it will take the same time.
The output voltage and input voltage ranges become very important with op amps that have a range that is something like +4v more than the negative rail and -4v less than the positive rail obviously because the range is very limited for lower voltage power supplies. They wont ever work right with a single supply of +5v only.
The other type which have their output go close to ground and not too close to the positive rail are still a problem sometimes with that single supply of +5v. That's because the output may be limited to 0.1 to 3.5v which sometimes isnt enough in a 5 volt system.