I will post considering IDLE conditions in 64 bit system.
Starting from system requirements , they actually Require the same as can be seen here for Ubuntu
The minimum memory requirement for Ubuntu 12.04 (32-bit) is 384 MB and
(64-bit) 512 MB of memory for Ubuntu
And for Cinnamon
x86 processor (Linux Mint 64-bit requires a 64-bit processor. Linux Mint 32-bit works on both 32-bit and 64-bit processors).
512 MB RAM (1GB recommended for a comfortable usage).
5 GB of disk space
Graphics card capable of 800×600 resolution
CD/DVD drive or USB port
Memory consumed by Compiz (Unity's Main ingredient) it takes upto 82~86 MB of RAM
Similarly for Cinnamon (w/o Desktop Effects), takes about the same amount as 82~86 MB RAM
Cinnamon (w/o Desktop Effects) in Idle state RAM usage
And for Unity it takes about 550~600 MB , the reason for that is i have Lots of Lenses installed hogging up-to ~15 MB RAM for each including Indicators. So that gets counted in extra usage.
With Desktop effects enabled in Cinnamon , it will take the same amount( 550~600 MB) or more . Well its only Traditional Desktop compared to Unity's Concept.
My Usage derived result says, there is exceptionally No Performance difference between them apart from Different Desktop Experience depending on
You should definitely give it a try to judge it personally.