You need a power supply that checks at least two conditions :
* regulated voltage
* voltage as stable as possible (very little ripple)
* isolation
* optionally, possibility to limit current
Regulated voltage means that the voltage won't change with the amount of current you take from the power supply.
regulation :
A computer power supply will be relatively regulated, as in the standard requires that the voltage must be +/- 5% around the voltage used.
A wall wart is usually not regulated - for example, a 7.5v DC wall wart may output 9-10v with just a led or your microcontroller connected to it, and may go down to 6-7v when go pull more current than it's designed.
It is possible to use a linear regulator to generate a lower voltage from the voltage generated by a computer power supply or a wall wart, and you'll get regulated voltage but a bit of ripple will still go through so this is not ideal.
Voltage stability (little ripple) :
Basically all computer power supplies and most wall warts are
switching power supplies. This means that in order to obtain 12v or 5v or whatever voltage it has to output, the power supply switches something on and off thousands of times a second sending pulses of power through a transformer and then on the other side, inductors and capacitors are used to smooth out those bumps of power, those pulses, into a straight DC voltage.
It's quite hard to make a power supply have no ripple or minimal ripple ... some wall warts have 300-500mV of ripple, computer power supplies can have 100-200mV ripple.
Just think of it like this: if you were to get a multimeter and make 1000 measurements of the output voltage within a second, you won't get the same output voltage more than a few times in a row, the ouput voltage will move around.
Here's an example from a cheap 30-40$ power supply and its 3.3v output :
The output voltage stays around 3.3v but it moves around between 3.2 and 3.4v which can be bad.
This ripple is not a big issue with microcontrollers but it can be if you use the power supply output as a voltage reference to measure something, or some things
Again, you can use a linear regulator to generate a lower voltage from the wall wart/power supply and reject some of that ripple, but some will still get through.
Isolation
Computer power supplies are referenced to ground. Some wall warts also are referenced to ground. That's not good if you want to use an oscilloscope on your circuit, you can blow it up if you test the wrong things.
Linear regulators don't help you here, proper lab power supply are isolated (or have an option for that)
Current limit :
Computer power supplies don't give you an easy way to limit the current going through the wires. If something bad happens in the circuit you make, the power supply can give you 200-500 watts or more. Cables can melt and burn up before a computer power supply will stop by itself in some cases.
Wallwarts can't give that much current so your circuit may not start burning, but some of them also don't have circuitry to protect themselves and can burn themselves up or fail sending higher voltage at the output.
So ideally you want a way to limit current in case something goes wrong in your circuit.
A linear regulator can help you there, as most have internal protections circuits that limit the maximum current they output to a value a bit above what they are designed to output and they will last for some time at that high value. But this is a bad idea in real world, you don't want to rely on it.
So conclusion... yes, wall warts are better than computer power supplies and are safer, but they still have quite a lot of drawbacks.
You also have to know how that wall wart behaves before actually using (is it regulated, is the ripple low, is it isolated or not?).
Search for an adjustable power supply with current limit option, they're a bit more expensive but they're a good investment.
Here's some examples:
http://www.circuitspecialists.eu/power-supplies/bench-power-supplies
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