Quick answer
aún (with accent) = still, yet — the same as todavía and swappable with it.
aun (no accent) = even, including — the same as incluso.
todavía only ever means still/yet, so it is the safe choice when you mean "still".
Side by side
Aún no lo sé. / Todavía no lo sé.I still do not know — fully interchangeable in this meaning.
Aun sin dinero, viajaron.Even without money they travelled — no accent, and todavía cannot replace it.
Aun así, gracias.Even so, thanks — a fixed phrase, always without the accent.
Memory trick
If you can swap in "even", write aun with no accent. If you can swap in "still" or "yet", both aún and todavía work — and todavía is always safe.
Typical mistakes
Todavía así, prefiero quedarme.
Aun así, prefiero quedarme.
The set phrase is aun así, "even so". Todavía cannot carry the meaning "even".
Aun no he comido.
Aún no he comido.
Here it means "not yet", so the accent is required. Without it the sentence reads as "even I have not eaten".
Check yourself
Fill in: "___ así, vale la pena intentarlo."