As Morlay said, try to pull the floppy cable off as well. You don't need it to power up the A1200; the boot procedure would only take longer. If the computer boot up fine now, there is a problem with the drive or the floppy cable.
You said that you weren't familiar with newer Amigas. From Kickstart2.0 and up you can access a menu by holding down both mousebuttons, either on boot up or on reset. See if you can access this menu. It should be displayable on your monitor.
If you say the A1200 will access the harddrive, my best bet right now would be that the previous owner have had a Multisync monitor, and therefore has set the screenmode to Multisync, which isn't displayable on a 1084.
Try the following
1. Hold down both mousebuttons on bootup
2. If the screen comes up, press "Boot with no startup-Sequence
3. Type "Devs:Monitors" (without the " of course) and hit Enter.
4. Type "Delete #?"
5. Reset. If my guess is right about the Multisync monitor, then you should be able to see a Workbench screen at this moment.
My final suggestion is to let the diskdrive be attached, put a floppy disk in the drive and power the computer up. If something happens, then it's okay, but if not, the mobo is broken I'm afraid.