This isn't so much a comparison, more of a discussion with myself about which one to focus on. First, the environment that you are in will determine this question much of the time - if you're on Apple, then the notes thing is built into their OS on desktops/laptops and into the iOS on your mobile devices. It has some nice features - encryption in transit, password protection, pictures and built into your iCloud experience. Here is where Keep has an advantage - it's available on nearly all platforms via the web browser. And it does most of those other things too.
Both companies are very clever. The interfaces, while different, share the same characteristics of note taking - different options for getting ideas down and into writing, while trying to make it all as straightforward as possible. They have in the main, quite slick interfaces too and very user friendly. We are truly spoiled for choice and this is part of the problem.
I use Macs, and PCs with Linux Mint and Microsoft Windows (in various flavours). I really like them all, but I prefer the Mac interface and hardware. Call me flash as a rat with a gold tooth, but it's a nice, neat and well put together combination. My main problem is - my personal phone is a Samsung S7 and work phone is an iPhone 6S so my personal notes are on the wrong device... but I like Apple Notes! I quite like Keep too, but it's too fancy for my liking. Yep - too many options and colours and other shit. I just need a piece of paper replacement and while both applications do that, I think Notes is tidier.
I like the sync across all devices I get from both apps - it's great and most helpful for keeping life in order, but this is where Keep shines - I can share amongst my accounts. That's pretty handy stuff that is. And Keep is in the cloud all the time - access via the web browser, apps on iPhone and Android now...
So which to use? I have a lot of data in both and the problem is, I know I've got something written down - like a username, but I'm buggered if I know where the damn thing is! 5 minutes of searching and I can find it. Pick one and stick with it I think. Keep seems the logical option - full sync across everything. But I prefer the interface to Apple Notes.
Which do you use?