A common misunderstanding by users is confusing a computer with a monitor. If one turns off the monitor you are not turning off the computer, so when you turn the monitor back on the computer will still be in the same state as before.
Another possible issue is that the power button on Macs can be set to act both as a sleep button and as an off/on button. If you only briefly press this button then it may put the computer to sleep and when you wake it then the computer will wake to the same state it was when you put it to sleep. To fully power it off you need to hold the power button on the computer for at least five second.
In the case of the Mac Pro it will have its own power button and display will also have its own power button. An iMac as comparison has a built in display and only has one power button which turns on/off both the computer and screen.
It is not clear which model Mac Pro you have, a current small round one, or the older silver tower. On the older silver tower one the power button is on the front, on the newer round one it is on the back where all the sockets are.