Budgie is fast. Mint is slow.
I use Budgie in an under-powered machine and it runs smooth. I also stopped distro hopping because Budgie looks good and I had zero problems as far as now (besides ibus which I fixed).
Just tried yesterday iso of last Mint/cinnamon release, I was pleasantly surprised. I started with mint 10 years ago (Ubuntu 9.04 I guess), and I have to admit that it is a very accomplished DE. But hey, I donāt know why, but I prefer Budgie. 
I installed Mint-MATE on a friendās computer several years ago and she is very impressed with it. I tried Mint-Cinnamon and liked the look of it, but the menu was ridiculously slow and there were other long-standing bugbears which noone seemed committed to tackle.
With Budgie things work, but if you do need help there is a commitment to fix it, although perhaps sometimes a tendency to blame an application. Iāve found with other distros that there arenāt really enough people involved to get an answer on their own forum, so I have used LinuxQuestions, but with UB you get help quickly. The people are also genuinely helpful - no āgurusā who donāt read your question properly, then give a quick and irrelevant answer just to increase their score/rating. No ignorant/insolent/impolite people either.
You will gather that second to the basic/important stuff working, the most important thing to me is the type of people involved 
Third is configurability of the desktop - a Budgie speciality.
I am currently using Linux Mint 19.1
Ubuntu Budgie is my backup, in case Linux Mint slows down for some reason.
Cinnamon has been around longer but it doesnāt reflect on stability or performance for me.
Regardless of the distribution, Cinnamon has always had UI lag, mainly the panel animations and extensions/applets. Budgie, certainly, the FNG of Desktop environments lacks some features but the basics are already baked in or easily added with an applet and performance as well as stability has not disappointed me one bit.
Looks wise, Cinnamon has always felt older, nowhere near as old as MATE though, while Budgie feels like a breath of fresh air.
As for UI setups, I consider Budgie to be better. I can set it up a bit like GNOME 2 and still feel like a modern UI. The Global menu and pixel saver applets do wonders but the desktop is also CSD oriented in some areas that really grind my gears. Cinnamon comes with non CSD apps but it lacks proper support from the developers to have a global menu on panels.
Current window decorations in Cinnamon need an overhaul, perhaps a proper theme would suffice, panel theming and its applets need a slight redesign too. I think Budgie should make its own mind. It can be friendly towards both CSD and non CSD but it canāt be both at the same time as it stands right now.