Is a simile closely related to an allusion?

A simile is not closely related to an allusion.  A simile is a comparison of one thing with another, linked with the words like or as.  For example, my love is like a red, red rose compares the speaker’s love to a rose, and uses the word like to facilitate the comparison.  Calling someone slippery as an eel is also an example of a simile, comparing someone to an eel using the word as.

An allusion, however, is simply an indirect reference to something or someone.  Mentioning fire and brimstone or the walls of Jericho in a story, for example, would be making Biblical allusions.  So, while a simile is a direct comparison of one thing to another, an allusion is simply a passing reference or mention.  An allusion could be a part of a simile, as in The underdog won the match like David against Goliath, but in such an instance we would have two completely separate devices at play – one a comparison, the other a reference.