There is nothing "special" about an aircraft engine that requires lead in the fuel -- Engines don't much care about the lead in tetraethyl lead, nor do they much care for it (it gets everywhere, fouling sparkplugs and contaminating the oil). What they care about is the octane (anti-knock) rating of the fuel.
Many "low performance" aircraft engines run just fine on unleaded fuels with a lower octane rating (among them the Lycoming O-320 and O-360 family that power a good chunk of the GA fleet), but high-performance aircraft engines (e.g. the IO-520 you'll find in Beech Bonanzas) require a 100 octane fuel. The approved specification for 100 octane aviation gasoline (ASTM D910) lists tetraethyl lead as the octane booster of choice.
It's also worth noting that some of the octane boosting techniques used in automotive gasoline are not acceptable for aircraft engines (the most common in the US being the addition of ethanol, which has two undesirable effects: reducing the energy content of the fuel, and damaging aviation fuel system seals and other components).
So why do we still make 100LL, and why don't we offer the other unleaded options at every airport?
Aviation gasoline is a minuscule slice of the gasoline market, so it doesn't make sense to have 5 tanks with 5 different grades of fuel at every airport:
All aircraft engines will run on 100LL, so 100LL (leaded) aviation gasoline is still produced because it is a "single fuel" solution to piston aviation's needs.
If leaded avgas were to disappear tomorrow the engines that require it would be left without an approved fuel, which would result in those aircraft being grounded until such time as an alternative fuel could be developed or the engine manufacturers develop a procedure for derating the engines (operating them at reduced power). For obvious reasons neither of these options is attractive, particularly to folks who own higher-power engines which were presumably purchased for the performance...
Changing the avgas specification is quite a bit of work - it requires ensuring that the new fuel is a "drop-in replacement" for 100LL -- one which can be mixed with 100LL in any proportion, and will work correctly in any engine designed for 100LL fuel.
The FAA is working with engine manufacturers and major fuel producers on a program which will result in such a specification (a Google search for
unleaded avgas transition also produces useful results), and hopefully a universal unleaded aviation fuel specification and one or more products will come out of those efforts.