Separate SSD disk for faster gaming?

Unless your 120 SSD is really better than your 240 one (if the 120 has a SATA version higher than the 240 for example and your motherboard support it), you won't notice a difference if there's any. It might be a little bit faster under some circumstances.

The main advantage of having a different SSD for your games will be that it won't be accessed when your OS needs to access its own files. When the OS is loaded and its resources are also, it doesn't read/write a lot.

If there is performance difference having your games on your 120 SSD, it will be when your OS also needs to access its own files. That is considering both are on the same SATA version.

If you want the most out of your SSDs, use the one with the fastest read/write speed on small files for your OS, the other for your games.

Since you're already on SSD, adding another as an extra drive will not improve performance much (if at all). The difference would be negligible unless you had a really bad drive or one that already has bad sectors. Any SSD would be faster than a traditional hard drive.

With that said, you should still consider using it as a game drive instead of using your main OS drive. By doing so, you can prolong the life of your main drive as you're doing less reads/writes to it when playing your games. SSDs won't last forever so you wouldn't want to kill it off too quickly.

If you wanted to actually improve your performance with the extra drive, you'll want to look into setting up your drives in a RAID configuration (e.g., RAID 0). Just be warned that by doing so, depending on the level you choose, you may increase the chances of potentially catastrophic failure.

And if you didn't get the hint, make sure you have backups in any case. Consider a regular high capacity HDD for backups if you don't already have one.

If the two drives are on different datapaths, you will notice a substantial improvement. Most motherboards have all drives on the same datapath though (some high end ones have two separate datapaths). If you are looking for an improvement with only a single datapath, you have to find what the maximum thouroughput and average thouroughput of the path is, and see if the single drive is "maxing out" that path. If it's not, another drive will allow faster reading. With SSD's though, you almost certainly have maxed out the datapath, and will not see improvements.