What is the difference between nachvollziehen and verstehen? In my dictionary they both mean “to comprehend”. Do they have a big difference in usage?
If you are either able to retrace or to reconstruct something or able to follow someone, then you can use nachvollziehen. If, however, it's just about understanding and comprehension of the nature of something, you can only use verstehen.
Both words have an overlapping sense but verstehen is a little broader in meaning. You can use verstehen when you are able to hear someone, as in Can you hear me?:
Or if you didn't understand someone, then you say:
Apart from that, verstehen is basically used in the sense of to understand the meaning or significance of something. Examples:
Nachvollziehen is synonymous to verstehen in many contexts. I think there's a simple way to find out if that is true. This is actually already mentioned in the other answer, I will just put it in another way again.
The meaning of nachvollziehen is to retrace or to reconstruct something in mind. Example: If you are able to retrace the feelings someone has when someone close to them died, you can say1:
Another example: Your math teacher explains how to solve an equation and you are not able to follow him, i.e. how to get from, e.g., step 3 to step 4, you can say:
In both example you can still go with verstehen though.
1: I am aware that you hardly ever will be able to fully understand what someone is feeling in such a moment; it's just an example to make my point.
"Nachvollziehen" and "verstehen" both are very similar in meaning (as you already found out), but there are some subtle differences that I want to emphasize here.
These details are kinda specific, so you should be fine with using both words synonymously in (most?) other places. When in doubt, use "verstehen" since it's more common.