For me personally, I would say that Western cartoons USED to be better than anime, although the quality in shows has declined noticeably since the early 2000s. Yes, I know that sounds like a very "90s kid" thing to say, but I just don't find modern cartoons like Steven Universe and all the rest it. It's just not for me. However, there are plenty of anime out there I've not seen yet that I enjoy, and they're making new ones all the time that I want to see and enjoy watching. I look at modern day cartoons, I might watch a few episodes, but I'm not motivated to really watch any of them; they bore me.
Cartoons from my youth just keep on giving, though, especially the Warner Bros. ones. As a kid, I loved the antics of the Animaniacs, Pinky and the Brain, etc. Now that I'm older, I love the dialogue, the parodies, the references; all the stuff that went completely over my head as a kid. From being one of my most loathed cartoons, Taz in Tasmania has become one of my favourites, and I have a whole new appreciation for Freakazoid as well. That kind of humour isn't present in modern cartoons OR anime, and I will always love it for that. Similarly, superhero cartoons like 90s Batman, Spiderman and X-Men series are substantially better than their modern-day counterparts. Well, Brave and the Bold was fantastic, but the new series of Marvel cartoons are all atrocious.
I don't feel like Western cartoons really cater to me these days; my tastes have changed as I've gotten older, and they don't make a cartoon for that...at least not yet. Anime, however, does; I might have to hunt about a bit to find something that appeals to me, but I CAN find something, which is my point. Maybe it's a case of quantity: there are a lot more anime than cartoons these days, after all. Plus stuff has gotten worse as time has gone on...Winx Club is utterly unwatchable now, for example.
...also, there's too much CGI going on these days. The new Thunderbirds is TERRIBLE. The Captain Scarlet CGI they aired a few years ago was pretty good, but they really dropped the ball with this one. I shudder to think what's going to happen if they adapt Stingray next; I loved that show. The new TMNT cartoon is a far cry from the stylish, vaguely cel-shaded series they aired before that as well, at least before that godawful Fast Forward season. There is an increasing use of CGI in anime too, and it bothers me - it's often not particularly appealing visually, and quite jarring when shown alongside traditional animation.
I still have trouble considering the two mediums as separate sometimes though...partly because anime was aired right alongside cartoons when I was younger, and it wasn't until I was older than I found out that anime and cartoons were separate. I don't really care where it comes from, as long as it entertains me. It's just most (read: all) of what entertains me these days comes from Japan.