From what I've encountered in spoken and written German so far, I can see two (and a half) cases:

Literal meaning
=
"auf jeden Fall" can mean literally "in any case" or "always".

> Auf jeden Fall musst du nach links und rechts schauen, bevor du über die Straße gehst.
 
meaning

> In any case you have to look left and right before crossing the street.

Filler word
=
"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][1]):

> Jedenfalls bin ich dann früher nach Hause.
>
> Auf jeden Fall bin ich dann früher nach Hause.

meaning

> Anyhow, I went home early.

"Auf jeden Fall" can be used literally and as filling word, "jedenfalls" only as latter.

Modal particle
=

The word "jedenfalls" can shift the focus a bit, often restricting a statement in the sense of "at least"

> Ich war (jedenfalls) pünktlich zuhause.
>
> I, at least, was home on time. [But I can't speak for others.]

or

> Das hast du (jedenfalls) früher immer gemacht.
>
> (At least,) you used to do that. [But maybe/apparently not anymore.]

or

> So scheint es jedenfalls.
>
> Or so it seems.



**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:

> 1: an Vorangegangenes anknüpfend: auf alle Fälle, bestimmt
> 
> 2: an Vorangegangenes anknüpfend: nach, trotz dem, was vorher geschehen ist

which I'd translate to

> 1: following something previous: in all cases, surely
> 
> 2: following something previous: after, despite what happened before


See the following example sentence from http://de.thefreedictionary.com/jedenfalls

> Ich weiß nicht warum, aber jedenfalls hat sie ihn verlassen.
>
> I don't know why, but she left him.

To me, this sentence doesn't change its meaning if you omit "jedenfalls":

> Ich weiß nicht warum, aber sie hat ihn verlassen.



[This site][2] lists many fillers and "jedenfalls" is even part of a [list of fillers to avoid][3]

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.


  [1]: https://www.schreiblabor.com/orthography/filler/
  [2]: https://www.schreiblabor.com/orthography/filler/
  [3]: http://juttas-schreibtipps.blogspot.de/2011/09/giftliste-iv-noch-mehr-fullworter.html