Abhorrent describes something truly horrible like finding a dead rat in your soup, but something aberrant is just abnormal, like a cat in a pink fedora.

Abhorrent means disgusting or detestable. When you abhor something, you loathe it. Its Latin root means, "to shudder, recoil," which is what you do when faced with something abhorrent like a zombie, or in these examples:  

She said sanitary conditions in the emergency room were abhorrent. (Seattle Times)

If I thought monarchy abhorrent, for example, I would not seek British citizenship. (Washington Times)

Aberrant (no "h") means unusual, straying from a defined path. It's not necessarily nasty. It's from the Latin, aberrantem, "wandering away." It's related to err, a mistake, through Old French and Latin. If a behavior is aberrant, it's just not normal:

Weeks ago the nation witnessed the culmination of aberrant fan behavior when Bryan Stow was beaten in the Dodger Stadium parking lot. (Newsweek)

Upstairs the show gives way to delicious aberrant moments, like the gallery kitted out in beige walls and chunky red molding. (New York Times)

If you can remember that the err in aberrant indicates an error and that you abhor something that is abhorrent, your word choice will be neither aberrant nor abhorrent. Wearing a feather boa to a funeral? Aberrant. Stealing the body? Abhorrent.