Not a teacher, but an IT Technician.
Your computer hangs (or freezes) when it stops responding to any inputs (mouse, keyboard). You can see everything on the screen, but your computer does not react. It is usually a software problem.
Your system crashes when the operating system has to deal with a hardware exception that can't be handled. In Windows it results in the infamous Blue Screen of Death. It can be either software or hardware problem.
Your computer crashes when one of its hardware elements breaks down. Depending on what breaks down, the computer may stop working or it may work with regular system crashes.
But it really doesn't matter because most people use the term "crash" for all events described above. ("My computer crashed.")
Btw. I have never heard/seen "hangs/hanging about" except for "My computer hangs about five times a day." If it is working slowly, people usually say "My computer is running slow."