OFF-TOPIC because it's about a different word, wobble, but Su.Ki.'s comment brought it to mind. This is a quotation from the British humorist, Jerome K. Jerome, from Three Men on the Bummel, about a bicycle tour through Germany in the early 1900s:

He said: “This front wheel wobbles.”

I said: “It doesn’t if you don’t wobble it.” 

It's a kind of joke because the first speaker uses wobble in the usual, intransitive way--he's suggesting the wheel is dangerously loose and needs to be fixed. The second speaker uses it in the rarer, transitive way--he's suggesting that the wheel is perfectly safe because the wobbling only happens "when you wobble it," i.e. apply force to make it wobble. The first speaker foolishly tries to fix the wheel anyway, and makes it worse.

Another difference is that 'wriggle' is generally intransitive and has no object, while 'wiggle' is usually transitive and takes a direct object.

'Keep still! Stop wriggling.'

'That lock's rather stiff - try wiggling the key a bit.'

Both words imply a back-and-forth motion, often of the body.

Wriggling is a specific kind of wiggling. Wriggling implies some kind of wave-like motion of the whole body. Wriggling suggests forward progress. Snakes wriggle. Worms wriggle. Eels wriggle. Someone exploring a cave wriggles their way through a tight space. A dancer who makes graceful motions that seem to flow like a wave across his or her whole body can be said to be wriggling, although it's not complimentary.

Wiggle just means a back-and-forth movement of something or some part of one's body. It usually means a small movement. For example, I can wiggle my ears. They just move up and down. We usually talk about a dog's tail as "wagging," but a pig's tail is much smaller and we might talk about a pig's tail as "wiggling."

Someone might say "when you flush our toilet, if it doesn't stop flushing trying wiggling the handle."