If you really want advantages of the Note 8, which is what you asked for, then I'd personally say that the S-Pen and microSD expansion are the obvious ones. The S-Pen is the Note series' USP, so if you don't care about that then I'd question why even a Samsung fan would want the Note 8: the S8+ is very similar and a lot cheaper. There's also a difference in screen resolution, but if the Mate's screen looks fine to you then you shouldn't care about that either (and lower resolution means less work for the gpu to do, which improves battery life).
SoC comparisons depend on which country you are in, since Samsung use Qualcomm processors in North America and their own Exynos in most of the world. The biggest difference to my mind is the gpu, where the Qualcomm Snapdragon 835's gpu is much more power-efficient than either the Exynos or the Kirin 970 used by the Mate 10 (though the 970 is much better than the Kirin 960 used by the Mate 9, which was frankly dreadful). If you go in for long gaming sessions this might be relevant, since it means it's more likely to heat up and throttle (something short runs of benchmark apps won't tell you). If that's not your thing you probably won't notice. In my opinion the responsiveness of the user interface is far more important than any benchmark scores.
Cameras: have you looked at pictures taken by both (real world, not test charts) and decided you are happy with the Mate 10? If so then that's fine, don't listen to anyone else because the only thing that matters is whether you are happy with them. If you've only looked at specs, then those tell you nothing at all about image quality and you should look at some real images - image processing is the single most important thing in smartphone photography, and specs tell you nothing about that.
The other obvious difference is the user interface. Both Samsung and Huawei put their own skins over Android, and both are quite different from stock. Some people like one or other better, some don't care.
I don't offhand know how good Huawei are with software updates. Samsung will provide a couple of years' support for the Note, and are I think generally quite good with security updates but can be quite slow with OS updates (8.0 for the S8 is still in beta I believe). Again, this matters a lot to some people and very little to others. Personally I think it's something people often make too much of a fuss about, but as I own a Pixel now it's very easy for me to say that
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I should add that I've no particular bias here: I don't own either phone, as both are too large for me (though I did play with a Mate 10 Pro the other day - it wasn't powered, so I couldn't assess the UI responsiveness, but it felt well made)
