MG15 is louder with a fuller low end and a nicer clean tone. However, Katana mini also has a huge volume which is more than enough for bedroom practice. Katana mini has a bit more gain for heavier playing, but the gain on MG15 is sufficient for a lot of playing styles.
Katana mini is much smaller and can be powered by batteries, although 6 AA batteries last for only 7 hours. It's also cheaper and comes with a delay that can be set as a reverb.

Overall, if you have plenty of space, you can play loud, and you play more clean to mid-gain stuff, Marshall MG15's louder volume and fuller clean tone may suit you better. On the other hand, if you have limited space, you have to lower the volume, and you play heavier music more often, BOSS Katana mini may be the better choice for you.

00:00 MG15 clean-bridge
00:14 Katana mini clean-bridge
00:28 MG15 clean-neck
00:43 Katana mini clean-neck
00:59 MG15 clean VOL MAX
01:23 Katana mini clean VOL MAX
01:45 MG15 crunch
02:23 Katana mini crunch
03:07 MG15 gain MAX
03:41 Katana mini gain MAX