We tend to think of the second one as "back of the evenlope" estimation, which is both pervasive and important in software engineering and related fields (eg capacity planning, resource allocation, etc). I wouldn't necessarily attribute this to mental math however. You can still be good at this without mental math (IMHO).
I might rephrase it to something like: mental math is a reflection your ability to use numbers as one would use tools. So, for example, when you learn probability you use lots of examples of games of chance or you boil it does to coin tosses, drawing stones from bags or rolling dice. Part of doing this is being able to (among other things) correctly negate probabilities, which tends to be far more obvious if you have a foundation in set theory.
Example: expected values. If you roll a d6 you expect to roll a 1 approximately once in every six rools. You get an array of possibilites from this based on how you choose to look at it (eg the probability of rolling a 1 after N rolls or the probability distribution of M 1s from N rolls). A common question comes up is "what is the probability I don't roll a 1 in N rolls?" and those naive in probability will often get this wrong. The answer is of course 1-(5/6)^N. And while you do learn that in probability, even if you don't specifically learn it I find that people who are comfortable with numbers as tools (as evidenced by mental math) will tend to figure it out anyway, or at least a good approximation of it.
Edit: corrected "not rolling" to "rolling". My bad.
To see that your answer for “what is the probability I don't roll a 1 in N rolls?” is incorrect, consider what happens as N gets large. The probability → 1, which should be intuitively wrong.
I might rephrase it to something like: mental math is a reflection your ability to use numbers as one would use tools. So, for example, when you learn probability you use lots of examples of games of chance or you boil it does to coin tosses, drawing stones from bags or rolling dice. Part of doing this is being able to (among other things) correctly negate probabilities, which tends to be far more obvious if you have a foundation in set theory.
Example: expected values. If you roll a d6 you expect to roll a 1 approximately once in every six rools. You get an array of possibilites from this based on how you choose to look at it (eg the probability of rolling a 1 after N rolls or the probability distribution of M 1s from N rolls). A common question comes up is "what is the probability I don't roll a 1 in N rolls?" and those naive in probability will often get this wrong. The answer is of course 1-(5/6)^N. And while you do learn that in probability, even if you don't specifically learn it I find that people who are comfortable with numbers as tools (as evidenced by mental math) will tend to figure it out anyway, or at least a good approximation of it.
Edit: corrected "not rolling" to "rolling". My bad.