This is actually a really hard question to answer, as there are a ton of factors.
I'll try to cover a few.
Visibility is probably the reason you most hear when it comes to GA pilots.
High-wing aircraft offer better visibility below the aircraft, especially for passengers in 4-seat or larger aircraft, as the wing doesn't block it.
Low-wing aircraft can offer better visibility above the aircraft, as the wing remains mostly out of the field of view.
Note that a lot of midair collisions involve mixed types; low-wing aircraft descend into high-wing, or high-wing aircraft climb into low-wing.
Landing characteristics are different, as low-wing aircraft can incur more ground effect than high-wing.
Aerodynamic stability of the two designs differs a bit; the center of lift can be higher in a high-wing design, often offset with dihedral in low-wing aircraft.
Ground clearance (and debris protection), specifically in multi-engine aircraft, is going to be greater when engines are mounted on a high wing rather than a low wing.
This is true of both propeller and jet engines, although propellers tend to have a larger radius for a given aircraft size.
There are a bunch more, but that ought to get the argument discussion going.
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@Egid, your already gave really good and correct answer dealing with the most important benefits and downsides, or simple the most significant differences.
I like to follow your invitation and add the differences I thought of and you have not already mentioned:
The additional ground clearance also benefits the maximum crosswind some high-wing aircraft can cope with, what in my eyes is also an important point.
Also high wing aircraft - talking about light sports - benefit the design of the fuel system as it allows consumption from both tanks simultaneously without installation of additional fuel pumps
- this is not only a benefit for lazy pilots but has probably already prevented some accidents.
Now somebody needs to tell me why I'm flying low-wing ;)
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Low wing aircraft also benefit that in a crash, much of the crash energy can go into the wings instead of into the fuselage.
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Low wing aircraft have the advantage in a water landing because the wings can keep much of the fuselage above water, as well as provide a temporary staging area for deplaning passengers.
In a high aircraft, you would be lucky to get out before drowning.
Likewise, on land, low wing aircraft are easier to control in a wheels up landing.
Low wing aircraft can be designed with a more widely spaced and shorter main gear, which is more stable.
Most high wing aircraft use fuselage sponsons (MU2, ATR-42/72, Do228/328, BAe-146/Avro RJ, C-130) to house the main gear, providing a relatively narrow, and therefore, less stable track.
For those high wing designs that house the main gear in wing-mounted engine nacelles or in the wings themselves (Twin Commander, F-27/227, Dash 7/8, B-24/32), the gear must be made much longer, and therefore, much heavier.
A long main gear design is also more prone to structural failure in a hard or poorly executed crosswind landing.
Low wing aircraft are easier to refuel, and if equipped with wing-mounted engines, those would be easier to service.
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Here are a couple of ground comfort related differences.
When it's hot, and you're waiting in line for fuel, or if your plane isn't hangared, you can stand in the shade under a high wing.
If you want to walk from fore of the wing to aft of the wing, to get the oil rag you forgot, with a high wing, you duck a bit and walk straight, with a low wing, you walk all the way around the wing.
Twice.
aviation.stackexchange.com/users/12116/Greg Taylor
Another factor is fuel flow (at least in small GA planes).
Lets compare a Cessna 172 to a Piper Cherokee.
The Piper is low wing and requires a fuel pump (engine driven with an electric backup) to flow the gas from the wing tanks up to the engine.
This add not only adds another part that can fail but the possibility that you can vapor lock the system if you run it dry.
In a the Cessna (high wing) the fuel is pulled by gravity, out of the wing tanks, and into the engine much like an old car (1930's era).
This eliminates the fuel pump and the vapor lock issue which some people find quite nice.
At this point I am well into the habit of switching tanks every 30 minutes.
A note should be made that some Cessnas (from what I have read) do have fuel pumps now
but I don't fly them
so I don't know a lot about the newer model specifics.
High wing airplanes also prove to be advantageous if you are flying a lot of back country stuff.
The added clearance can help you land in bushy areas or other less hospitable places where a low wing plane might strike something.
The best answer to this question can usually be found at your local airfield.
Ask a guy that owns a Piper (just not a Cub), then ask a guy that owns a Cessna (just not the 400), and decide for your self based on their answers.
aviation.stackexchange.com/users/3677/Dave
Another factor with low-wings is that on take off and landing, the low wing will block airflow over the empennage, reducing rudder and elevator effectiveness.
That's why so many low-wings have T-tails, to keep the tail in the flow of air during landing and takeoff.
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A quick addition to the previous answers:
Larger commercial aircraft also capitalize on the low-wing's benefits from a maintenance perspective.
aviation.stackexchange.com/users/2934/Vanessa G
Let me add one more advantage/disadvantage pair.
High-wing aircraft (at least for small GA aircraft) may be easier to preflight as wing fuel drains are more accessible and it is easier to examine the underside of the wing (flap linkages, aileron hinges, skin, possible fuel leakages).
Personally, I'm tall enough to be able to see the top of the wing without a ladder, but I'll admit that this advantage shrinks for shorter pilots.
The downside is ease of refueling (and sticking the fuel tanks).
A Cherokee (low-wing) doesn't require a ladder to refuel whereas we carry a stepladder in our club Cessnas.
aviation.stackexchange.com/users/3211/Peter Hansen
What are the advantages and disadvantages of a high-wing vs. low-wing aircraft design? When might one prefer one to the other?  Is the answer the same for large and small aircraft?
What are the pros and cons of high-wing compared to low-wing design?
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There are a lot of different factors. Visibility  is probably the reason you most hear when it comes to GA pilots. High-wing aircraft offer better visibility below the aircraft, while low wing aircraft have the advantage in a water landing because the wings can keep much of the fuselage above water. While this is a hard question to answer due to the factors involved, it is generally agreed that high wing aircraft enjoy a little more visibility towards the ground, a little more ground clearance, less ground effect, less cross wind and offer a place to take shelter from sun in hot climates. On the other hand, low wing aircraft have an advantage in crash landings as much of the force goes into the wings instead of the fuselage, and have better visibility above. 
Disclaimer High wing aircraft characteristics Low wing aircraft characteristics