Do the words imbalanced and unbalanced mean the same thing?

Not exactly. The Merriam-Webster defines them as:

imbalanced- lack of balance; the state of being out of equilibrium or out of proportion

unbalanced- not balanced; not in equilibrium; mentally disordered or deranged; not adjusted so as to make credits equal to debits


http://www.merriam-webster.com/dictionary/

Imbalanced :

A lack of balance, as in distribution or functioning.

Unbalanced:

1. not balanced or not properly balanced.
2. lacking steadiness and soundness of judgment.
3. mentally disordered; deranged

http://dictionary.reference.com/browse/unbalanced?qsrc=2888

Imbalanced lists Unbalanced as a synonym but Unbalanced doesn't list imbalanced.

Unbalanced is used more for an emotive state while imbalanced is more physical.

As adjectives, they mean principally the same thing:

If you describe a situation as imbalanced, you mean that the elements within it are not evenly or fairly arranged.

un·bal·anced (n-blnst) adj. -- Not in balance or in proper balance.

http://dictionary.reverso.net/english-cobuild/imbalanced

http://www.thefreedictionary.com/unbalanced