If you're using the word 'cartoon' to mean an animated show/series/thing/whatever that involves drawings moving around, then yes, anime are cartoons just like everything else of that nature. For the most part, you could lump all similar things together on the basis that they have cartoon style; i.e. the drawings are stylised representations of things. I don't think the word cartoon is inherently offensive, and I'd like to take this moment to point out that there are Japanese animated series that are every bit as 'childish' as some western ones. Probably because they're, you know, *made for kids*.
However. the Japanese have leant towards tackling more overtly adult themes in an animated format, and as such if the word 'cartoon' conjures up images of kids shows, then you're going to take exception to it.
My view is that some anime are 'japanese cartoons', just like western ones except with a slightly different aesthetic and a Japanese soundtrack.
Other anime are somewhat different, and it's easier to refer to them as 'anime' to distinguish them from the more kid-oriented ones.
What I think people need to get over is assuming that because something is 'anime' that it's inherently better than a western 'cartoon'. It might be. It might not be. While I acknowledge the Japanese willingness to tackle more adult themes in their animation, I also point to the tide of overwrought, emo, painfully adolescent tripe that comes with it. Just like I acknowledge western 'childrens'' cartoons for their ability to subtly refer to adult themes in a way that goes over the heads of the kids but makes an enjoyable 'other layer' for adults, while pointing to the fact that western animation has yet to become really comfortable with overtly 'grown-up' themes.