Unless your 120 SSD is really better than your 240 one (if the 120 has a SATA version higher than the 240 for example), 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.

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.