Probability: 6 Dice are rolled. Which is more likely, that you get exactly one 6, or that you get 6 different numbers?

The question is very easy to answer without computing probabilities. Every combination with six different numbers contains exactly one six. There are then additional combinations which contain exactly one six - e.g. $111116$. So the probability of exactly one six is greater.

The number of rolls with exactly one 6 is $$\binom{6}{1}5^5=18750$$ the number of rolls with all dice different is $$6!=720$$

For $5$ dice, the number of rolls with exactly one 6 is $$\binom{5}{1}5^4=3125$$ and the number of rolls with all dice different is $$6\cdot 5\cdot 4\cdot 3\cdot 2=720$$ so the number with exactly one 6 is still larger.

then it is more likely that you will roll exactly one six.

Intuitively makes sense, because in each of the combinations where every dice is different, there is exactly one six. Therefore, there are more cases of one six than all different.