1) To a computer the number line is not solid like they draw it in mathematics. Instead it looks like the dots and dashes of morse code. This is because some numbers do not exist for a computer. For instance there is no 1 / 3 on its number line, just a big hole.
2) (a - b)(a+b)≠(a^2 - b^2). Algebraically equal expressions are not equal to a computer.
3) Addition is not commutative.
The order of addition can drastically affect the answer!
4) A computer can not subtract or multiply without possible disastrous error.
5) You never use the quadratic formula to solve a quadratic equation because on a computer it might give inaccurate results.
6) Newton's iteration though taught is rarely the best one for the job.
7) A computer can not always compare theoretically equal quantities.
8) There is a largest number to a computer.
9) A computer can have three states for a matrix, singular, non singular and nearly singular!
In mathematics, you don't understand things. You just get used to them.
I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.