How does the electron structure of the double bond in an alkene different from in an aromatic ?

Alkenes are bonded in a single line, while aromatics are molecules with six bonds in a circular shape.  Benzene is an example of an aromatic.

The double bond in an alkene is just that, a sigma bond and a pi bond between two particular carbon atoms.

In an aromatic system, you have alternating single and double bonds in the valence bond structure. In the specific case of benzene, you can write the double bonds between carbons 1,2; 3,4; 5,6 or between carbons 2,3; 4,5; 6,1. The real structure contains an equal mixture of both these possibilities. You have resonance stabilisation, so all the bonds are equivalent, and while the apparent valence bond order is 1.5, the actual bond strength is actually a bit greater than that implies.