I currently run Linux Mint on a USB Flash drive (16GB) to an Intel Celeron NUC. I only use it for browsing and VLC Media playback - it's ok but not exactly snappy.
I was therefore wondering whether installing Linux Mint on a 120GB Solid State drive connected via one of these would make the experience any better? Or not?
Without knowing anything about the drive or the rest of the system it's very hard to give a definitive answer. If it's a high end thumb drive I wouldn't expect it to make much difference. If it's a mid-range one it may or may not help depending on what the rest of the system is and if it's just a cheap one then it should help even if the rest of the system has limitations too.
Have you actually installed Mint to that drive or are you running it as a Live disk?
Also, why not use the SATA port inside the NUC instead of buying an external adapter?