Why not put many watts through a mag mount ?
For one, magnet mounts use capacitive coupling, which means there is a space between the magnet and the metal it is "attached" to, effectively creating a capacitor, which means when there is enough power, just like a capacitor, there will be arcs through the dielectric (i.e. your car's paint). This will change the "dielectric constant" of the capacitor, which will affect things like the antenna's SWR, in some cases drastically. This can lead to blown final stage transistors, which in an amp can be a very expensive fix. Said capacitive coupling also tends to cause CMC problems on a coax. Have you seen or heard of anyone using coax length to "tune" an antenna? When this happens you are not tuning the antenna, your just using one problem (that many apparently don't realize is an actual problem) to hide another. This type of problem is especially common when using magnet mounts, and will be amplified by running a lot of power i.e. more than 100 watts) through said antenna system.
Why you use it if stationary ?
I'm not sure why he wouldn't. You do need a triple or quad magnet mount, a single five or six inch diameter magnet will definitely not be enough to hold that long of an antenna on a moving vehicle.
I am asking because I plan to go to a key down competition and I was thinking to use a full size 1/4 wave whip, just for the competition. Would it be more efficient than a center load coil antenna ?
A full length whip, through a magnet mount will be "more efficient" (hey look, those words again) than a center loaded whip, also through a magnet mount, and mounted in the same location.
The thing is, a proper permanent mount (like the Breedlove mount mentioned above) will me "more efficient" (hey look, now you've got me using those words...) than any magnet mount, and sometimes noticeably so. The reason is, again, the capacitive coupling in a magnet mount that I mentioned above, all of the RF energy that travels between the antenna and ground plane has to travel through a capacitor, which has its own impedance, and by the very nature of capacitors, will add loss.
The DB