Difference between "publicly" and "publically"

I know publically appears as an incorrect spelling in most dictionaries (in fact as I type this up on my Safari browser it keeps trying to correct the spelling to publicly).

However I have seen the word spelled in that manner before in certain places so I did a search on Google and found that there is a definition for the spelling publically at dictionary.com and the free dictionary as well. The English wiktionary site considers it an alternate spelling.

Are both spellings valid or are there some subtle differences I am missing? Which version would you accept as correct in a spelling bee?

Publicly is certainly more common, but publically is a logical alternative, mentioned in the OED with half a dozen citations from 1727 to 1998, so it can't really be called a mistake.

Adjectives ending in -ic form the adverb with -ally: poetic poetically, systematic systematically, automatic automatically.

There is one exception to this rule, public has publicly. And its only natural that speakers in the course of time extend the general rule to public and form publically. I would see this as a natural "pruning" and rectification of the system. Of course, there is no difference between the two forms. The first is the older irregular form, still preached by a lot of grammars, the second is the newer and regular form and in this case I must say sometimes speakers have more sense than grammars or grammarians.

Added: It should also be the task of dictionaries and grammars to regularize language and not to fix irregularities for all times.

I would accept only "publicly" as being correct. I'm surprised that you found dictionaries listing "publically" as anything other than a mis-spelling of "publicly".

If this alternative spelling does become commonplace, there's still no difference in their meaning; they are, after all, alternative spellings of the same word, not different-but-similar words.

Update: In the 2-and-a-half years since I wrote this answer it seems that "publically" continues to grow in popularity, and more dictionaries are listing it. Nevertheless I would still advise against its use since it is still not yet globally accepted as a correct spelling and there's nothing wrong with "publicly"!