What is the difference between empirical and theoretical probability?

Theoretical probabilities are your calculation of what
"should" happen, when you are able to describe all the equally likely outcomes.  Then
the theoretical probability is the number of "successes" (outcomes that fit the
probability you're trying to find) divided by the number of equally likely outcomes. 
For example, if you roll a fair six-sided die, there are six equally likely outcomes: 
1, 2, 3, 4, 5, and 6.  The probability that you roll an odd number is 3/6 (equals 1/2)
because three of those outcomes - 1, 3, and 5 - are odd, which makes them
"successes."

Empirical probabilities are your calculation
of what "did" happen in an experiment.  Again, you put the number of successes in the
numerator, and the denominator is the number of trials you did.  If I roll a fair
six-sided die ten times, one possible result is that I roll 4, 2, 5, 6, 3, 2, 3, 5, 2,
2.  My empirical probability of rolling an odd number in this case is 4/10 (equals
2/5). 

Notice that the empirical probability is not
necessarily equal to the theoretical probability.  Notice also that if you do the
experiment again, you may well get a different (but still correct) empirical
probability.  The theoretical probability, on the other hand, has only one correct
answer.

As for professions, consider industries like
insurance or casinos and name some of the people who work for those companies. 
Probability theory is also the foundation of statistical inference, used by
statisticians and researchers of all kinds.