First, let's mention that in order to consider these two expressions the same, you should think of « toujours est-il » as « il demeure que », which is more or less equivalent to « en dépit de tout ce qui précède », which makes the negative side of the expression stand out.
Then the two expressions might stand side-by-side a little closer in your mind.
Le problème, c'est que j’ignore toujours comment gérer cette situation... Toujours est-il que / Il n'empêche que j’en sais un peu plus, grâce à eux, sur ce qu'il manigance.
Rather positive : You're stating that though the situation is not easy and you cannot yet succesfully figure your way out of it, you’re staying positive because... (either)
... evolving in the right direction, because you know a little more than before about the situation.
J’en sais un peu plus, grâce à eux, sur ce qu'il manigance. Toujours est-il / Il n'empêche que j’ignore toujours comment gérer cette situation...
Rather negative : You're stating that yes, you know a little more about what’s going on, but... (either)
in spite of this new usefull knowledge, you don’t know how to deal with this situation.
this new usefull knowledge does not prevent you from being clueless on how to deal with this situation.
Let’s however mention that toujours est-il que could be interpreted as a simple statement of what is, a logical conclusion to what precedes it:
J’avais toujours eu un faible pour les roches, je collectionnais les cailloux depuis toujours et ne manquais jamais une occasion de demander une visite au musée de paléontologie pour mon anniversaire... Toujours est-il que je me lançai avec enthousiasme dans des études géologiques aussitôt mon école secondaire terminée.
...while il n’empêche que always implies a contrast with what precedes, something with which toujours est-il que is also quite comfortable with:
Je me suis rendu au rendez-vous sans attentes, imaginant bien que l’on se payait ma tête, mais toujours est-il / il n’empêche qu’une véritable passionnée d’astrophysique m’y attendait.
I cannot think of a context where il n’empêche que would fit while toujours est-il que would not... Sorry! It might only be me...
In a dialog
@Guillaume31 notes in a comment that we could think of these expressions as replies in a dialog, which is indeed a possible case. It is beside the original question, but it is possibly the best scenario to see the meaning of the two expressions clearly diverge (something the OP is rather clearly expecting). It also allows the illustration of some cases of common usage, and will show that...
- while « Ça n’empêche que... » does indicate an opposition (of things, concepts, opinions, etc.), it does not systematically indicate a disapproval and is not always confrontational; and that...
- « Toujours est-il que... » can very well be used in a not so positive interaction.
When someone comments what the other just said by simply stating « Ça n’empêche que... », I can think of two possible meanings, depending on the context:
- Confronting the other person: “You might have made some valid points, but I’ll stick to my version/opinion!”
- Supporting the other person against a state of facts being presented: “I agree with the opinion you expressed despite the opposition you described.”
On the other hand, when the reply is a simple « Toujours est-il que... », then again, two possible ways of interpreting it:
- Slightly annoyed or impatient: “Would you please get to the point!” or “Please skip of few steps, I don’t need this much details!”
- Sensing the conclusion is getting closer (especially if the conclusion was already mentioned, with justification following): “Now I see what you’re getting at!”
