Your new system will detect the old hard drive & should be able to access it fine as well.
However, as for just installing the drive & letting windows boot up on the new system - this may or may not work.
Booting it up in safe mode is almost guaranteed to work but booting normally may be a problem. It depends a bit on how similar the new system is to the old one.
You have to remember that the old windows install will be expecting a load of hardware that no longer exists on the new system, and the new system will have a load of hardware that windows doesn't have the drivers installed for.
Even if it does boot, you may find it's twitchy & unstable.
If you want to give it a go, I suggest you boot in safe mode first time round, go into device manager & delete all hardware that is not present in the new system. Then reboot normally.
Final line is, (unless theres loads of stuff on your old drive that you don't want to lose), I recommend a format & a fresh install every time