Snow Leopard and Mountain Lion are far apart from each other, and so seem to be the two versions of iPhoto you have.
Where did you get the iPhoto which is on your iMac? If from the App Store, you should be able to download and install it on the Mini as well by logging on to the App Store with your Apple ID and finding it under Purchases. Check?
After some research, the Mac Mini is Snow Leopard with iPhoto 9.2.3 installed and the Mac is Mountain Lion with iPhoto 9.4.2.
The Mac Mini is used for music/photo with my audio/video system in my great room for iTunes and iPhoto on the big sreen hdtv.
Since later versions of iPhoto (9.3,9.4 or later) require Mac OSX 10.7 or later, which the Mac Mini doesn't support, how do I get my pictures from the IMac to the Mac Mini for viewing?
I assume I could keep a version of iPhoto 9.2.3 installed on my external HD, but what happens when I copy them from the Mac to the external drive for use on the Mac Mini - they are still going to be in the latest iPhoto from my Mac and I still have the same problem - incompatibility.
The only way for you to have the same photos in these two different versions of iPhoto without an extra tool is to export them from one version, then to import them into the other. This may result in a lot of additional manual work to rearrange events and albums, etc.
A tool that should work for you is iPhoto Library Manager. You are within its minimum requirements for both OS X and iPhoto, the app is well supported and works flawlessly for me. All you need is having the two Macs on the same network - preferably a fast and stable one (wired Gigabit Ethernet, if you can) as IPLM will move a lot of data back and forth.
With IPLM, you'll only need an iPhoto Library on either Mac, the complete, maintained one on the iMac, and a new empty one on the Mini (created with the iPhoto version on the Mini). Then you "merge" them with IPLM. Check options an preferences carefully before you do so, though.
Later, as you apply changes or add photos to either library, you can merge them again easily. Again, double-check prefs carefully first as different settings than the first time might be required.