When to use whence instead of hence

I've come across instances where I felt using both was just fine. The dictionary definition doesn't provide much clarity either. Could someone please clarify the differences between the two?

As @tchrist comments, you need to consider the three sets of three:...

here, there, where;
hence, thence, whence;
hither, thither, whither

But in practice the last set are pretty dated/archaic forms in all contexts, and the middle set are normally only used metaphorically today.

Once you stop to think about the fact that the usage is metaphoric, it should become clear. Here are some written instances from Google Books...

([here is] some stated fact) hence we derive (some other fact) (i.e. - "from this fact here")
(some statement) thence we derive (some fact) (i.e. - "from that statement you just read there")
(some statement) whence we derive (some fact) (i.e. - whence=wherefrom)

You won't come across whence so often these days, but (as I hope those examples illustrate) it can sometimes be used in contexts where either or both the others would be perfectly acceptable.

Since the usages are all metaphoric, the "location" of the "statement" (or thing referenced by the statement) is somewhat uncertain. In speech/physical space, the difference between "There it is!" and "Here it is!" may simply depend on whether you're pointing a finger or spreading your hands as you speak.

But at any point within a written text, here could encompass the entire book you're reading, and there could mean just the previous sentence or clause. And where/wherefrom/whence can always refer to anything written previously (normally, the immediately-preceding statement).

To sum it all up - if you're not sure which to use, stick with the most common form (hence). If there's a strong sense of from there or from where in your context, use thence or whence.

Let's assume you know what here, there, and where mean.

Hence, thence and whence mean from here, from there and from where.

Hither, thither and whither mean to here, to there and to where.