Ethylene – π
In the most simple linear π systems, bonding to metals takes place by two interactions. Electron density is donated directly to the metal like a sigma bond would be formed. Also, the metal can donate electron density back to the linear π system (ethylene) from the metal's d orbital to the empty π* orbital of ethylene.
Allyl–π
Allyl groups can bond to metals as trihapto or monohapto ligands. Monohapto ligands bind mostly sigma orbitals and trihapto ligands bind using delocalized π orbitals. In essence the monohapto ligand binds the metal as an allyl group and the trihapto ligand binds over all three carbons, where the lowest energy π orbital donates electron density and the highest energy π orbital accepts electron density. 
The allyl complex is diverse because it can alter the metal's electron count by transferring between a monohapto (1 electron, η) and trihapto ligand (3 electrons, η). This fluctuation allows stability when a two-electron-donating group bonds or breaks from the metal.