It really depends on what you're looking for. The Kaiju eeks out a tiny bit more power, though certainly nothing you're going to notice. It's also a little bit quieter, but we have used noisier amplifiers on headphones before and not noticed that issue either, so that's not exactly a point in either favor.
The Kaiju is a bit less expensive but comes with lower quality parts. You could certainly upgrade those later on your own, and there is (barely) space to fit them in the amp. With the Kaiju, you will be married to the 300B output tube, which I don't think is a problem myself. I would say the Kaiju is a little easier to build overall. The form factor may also be more convenient for many, as it is a proper width to just take up one shelf on a stereo rack. The MonAmours are surprisingly large in comparison.
The MonAmours are a bit more expensive (we'll ignore the introductory sale thing) but offer film cathode bypass caps, some nicer coupling caps, and some nice cosmetic touches. The very deep low bass will be slightly better defined based on the altered operating point allowing us to use the full winding on the choke, but the Kaiju is already fantastic in this department. You will get the convenience of being able to use 2A3s right off the bat and the 2A3-40, which is really a 300B with its filament connected differently so it will run on 2.5V. I have been running the first pair for a few weeks now, and going from a Sovtek 2A3 to a JJ 2A3-40 and using the switch to adjust the current really does alter the amp from a 2A3 amp to a 300B amp sonically. There is also certainly the potential to modify the MonAmour to take 300Bs, and I hope to use the amp we built for the manual photo shoot on Sunday to perform these experiments.
I could definitely be happy with either amp, and I'm certainly satisfied having both around. My collection of 2A3s is far larger than my collection of 300Bs though, and some of those old stock 2A3s sound pretty nice. I also admit that I really like shrouds and meters. If cosmetics aren't much of a consideration and you're only ever going to run 300Bs, I'd stick with the Kaiju.