They are roughly synonymous. Both translate to at least in English. You can always say one when you can say the other, but depending on Situation one is preferred over the other.
If used as "as much or more than", mindestens is preferred:<pre>
Du brauchst mindestens 30 Punkte — You need at least 30 points.</pre>
But when it's not about numbers, wenigstens (or its synonym immerhin) is preferred:<pre>
Wenigstens sind wir noch am Leben — At least we're still alive.</pre>
It was posted in other comments already, but in case it helps anyone else looking here first, there it is again!
This site jerks all over the place, jumping back up to the top, over and over again, so it becomes unreadable! Has anyone else had this problem? I couldn't get past the section on what "-stens" means, so still don't know the difference in usage between "zumindest" and "wenigstens". (sigh) I was able to read that "wenigstens" is used with numbers.
In case you haven't found the answer yet (or anyone else is wondering the same): "an" is used before vowel sounds, regardless of whether there's actually a vowel. So you use "an" before an unvoiced H, but not before a voiced H: an honor, an hour, an honest mistake a horse, a house, a hammer
I may be wrong about this part, but I believe in some cases, it depends on the region, eg: US: a hero, a historic event UK: an hero, an historic event
"Wir brauchen mindestens eine Stunde." should definitely be accepted ("mindestens" is listed as the second meaning of wenigstens on Duden), and—to be honest—I'm pretty sure it already is, so I'd recommend not only reporting it if you come across it again, but taking a screenshot as well, as it might be a bug causing your solution to be rejected.
please can you explain
Just a little FYI: as you've not replied to a comment, that "you" is pretty ambiguous.