As the question says I'd like to know what the advantages and disadvantages of using a prop for magic spells are.
I'm specifically interested in the idea of throwing bean bags at people, and why it isn't done more often.
I've never LARPed, but I think I can still offer some input on this issue. If I'm totally out of place with my ideas then just disregard :)
I can imagine why using projectiles to represent spells isn't done very often. Using your beanbag example, it is very un-spell-like:
I think props could be used to represent spells, but you'd need something different than a thrown projectile.
For a fantasy setting where a magic user has a finite, replenishable resource than can be consumed to cast spells, I would recommend a super soaker where the water is your mana and you verbally communicate which spell you are casting.
For those with "memorized" spells that can only be used once, I'd suggest water balloons of different colors.
Obviously these suggestions are suited more toward outdoor fair-weather play. For indoor or colder settings there are air cannons that would work nicely.
Both of the systems I've LARPed in (NERO and SOLAR) use spell packets to represent spells, which are basically small bags of cloth filled with bird seed:
They have the following advantages over beanbags:
Typically when we cast a spell in these systems, a verbal component is required. Packets aren't labeled, they just represent the energy you throw after you recite the verbal.
There are numerous sites with instructions for making them, but I'll add a link to one of them here.
I think it depends on the form the LARP is taking. In the system I'm part of, we assume that the spells are mind affecting, therefore you can't dodge them (so no need for the bean bags) - BUT the spells can be resisted due to personal conditioning or personal heritage.
Some examples used in our system:
I assume the reason why bean bags are used is to keep track on the spells you have. In our system, mages have to prepare spell cards, which are ripped when a spell is cast. You learn to keep a tally in your head of what power you have left. But, since we don't have "magic missle" or "fireball" in game, we don't need projectile magic spells.
The style of LARP system you're part of would govern whether spell packets are required.
I've seen especially any kind of 'ball'-magic represented via props. F.e. a simple tennis-ball surrounded with a thick layer of silicone or thick cloth pieces filled with sand. Other magics are harder to simulate, but I've also heard about streamers and similar used.
Bean-bags (with only knowing what I heard about them in TV) do not sound like safe device to throw at people mid-battle. If you gonna throw something make sure that it can't hurt the player no matter where it hits, or make sure that the player knows what's coming his way (and he's fine with that).
To come back to your question, the big advantage is that you have something visual and physical players can react to. The big disadvantage is that you can run out of 'ammunition'.
This form of 'prop use' is actually found fairly frequently in 'boffer' LARPs. If I had to guess, its not used in 'no-contact' LARPs because of the aggressive action (and potential for injury) involved.