They mean the same thing, I think, but are used differently? When can I use one, and when can I use the other?
Jedenfalls is an adverb which slightly modifies the predicate of a sentence. It adds certainty and some kind of finality in its first sense [=definitely, in any case]. Additionally, it connects the statement to what has been said immediately before. Without any context jedenfalls doesn't make any sense.
Disregarding being a curious example, without any previous context this sentence is semantically nonsense. But adding the context you see that it's absolutely fine to use it that way.
In most cases you can replace jedenfalls with auf jeden Fall but not always. In the following sentence a more appropriate word to use is zumindest which accords with the second meaning given by Duden.
The expression auf jeden Fall is used in the same way as the first sense of jedenfalls. But you can use this expression as confirmation as well which is not possible with jedenfalls.
In respect to the other answer, I need to mention that both words are neither a filler nor a modal particle. Filler are words without any relevance to what is being said. And they are peculiar to spoken language as um or like in English.
Modal particles must fulfill a set of requirements to be a modal particle. There's one which fails for jedenfalls which also has been said by Emanuel in a comment:
To cut a long story short: Jedenfalls and auf jeden Fall do have one sense in common whose literal meaning is: "whatever happens or may have happened, this is true".
This is not an Answer that covers auf jeden Fall and jedenfalls as well as their differences and commonalities. It is not even an answer that covers all of jedenfalls: I merely wish to address one particular aspect of it.
In legal writing, you want to be precise and concise. You want to minimize ambiguity. To this end, it helps not only if you choose your words carefully and are clear about what you want to say. You also want to avoid being verbose and using non-functional language that opens a door for ambiguity to creep back in.
Hence there is no need for “filler words”, and if you see jedenfalls in legal language, you should assume it is there for a purpose. I shall demonstrate my contention using a collection of samples chosen by a leading search engine. (To narrow the search to legal writing only, I stuck in an abbreviation – BverfG for Germany's Federal Constitutional Court – that would not typically be found in other species of texts.)
I shall go through the search engine hits in order, leaving out only samples that evidently do not belong under the rubric of “legal writing”, such as journalism.
Does the meaning stay the same when we remove jedenfalls?
Absolutely not! The Court always checks first for procedural problems and whether the formal requirements for an appeal have been met. It found that parts of the appeal did not fulfill that criterion. However, that does not necessarily invalidate the appeal in its entirety. Next, it moved on to the “heart of the matter”, namely, whether the appellant had sufficiently substantiated her grievances. The Court found that she had not done so and hence, the appeal fails – irrespective of its partial failure on formal grounds earlier stated – in its entirety. Leaving out jedenfalls would omit this essential information.
Again I wish to quote Tonio Walter:
The Court found it important to note that this is not the only thing that cannot legally be demanded of husband and wife... but it wanted to leave open how far their protection against such demands extends, hence the jedenfalls. (As an aside, don't you marvel at the descriptive power packed by this one word? It sits there like a kernel of corn on a hot plate, then POP! twelvefold increase in size...)
This is straightforward enough. In English we would write “In any case” or perhaps even “Be that as it may”. Again, jedenfalls is essential and must not be ignored.
And so on, down the list of search results. I find not a single instance where jedenfalls could have been omitted as a “filler word”.
To sum up, I disagree with the first Answer to this Question.
I believe I have shown otherwise.
From what I've encountered in spoken and written German so far, I can see two (and a half) cases:
"auf jeden Fall" can mean literally "in any case" or "always".
"auf jeden Fall" and "jedenfalls" are sometimes used to start a sentence as a filling word. This is mostly used in spoken, but not in written language (see a list of fillers):
"Auf jeden Fall" can be used literally and as filling word, "jedenfalls" only as latter.
The word "jedenfalls" can shift the focus a bit, often restricting a statement in the sense of "at least"
Edit: In response to the comment that only suggested I'm wrong without specifiying exactly how, I'm adding a few sources and a third case I neglected.
http://synonyme.woxikon.de/synonyme/jedenfalls.php lists many words as synonyms for "jedenfalls" that have completely different meanings.
http://de.wiktionary.org/wiki/jedenfalls states two meanings:
which I'd translate to
See the following example sentence from http://de.thefreedictionary.com/jedenfalls
To me, this sentence doesn't change its meaning if you omit "jedenfalls":
Instead of downvoting and telling me I'm wrong, it would probably be more helpful for the author of the question if you tell me how I'm wrong or provide a better answer.