How to explain to en alien what's the difference. Alien in the meaning a person with first language other than English (*and possibly moonlighting, family members in burkas etc but that's another story)?

Well, first is milder, it advises and reminds not warns; it means not big deal if you forget.

The second warns and urges, alerts and admonishes; it means big trouble if you forget.

Sometimes the negative contraction of "don't forget" gets lost in translation or in auditory reception. If that happens the opposite meaning is received than was intended. "Remember" is more direct and a more positive request...

Actually the first one is stronger, as you can actually picture and imagine what remembering is. Anytime you put a negative into the mix, you can't really imagine not doing something.
This is from a visual linguist though. 😉

How about: "Don't forget to remember....!"