I don't find the use of the word "cartoons" offensive, when referring to Anime. Nor do i agree with OP's friends that people use it because they personally don't like Anime.
But the two words mean different thing, and it is not only confusing, but simply incorrect to refer to Anime as "cartoons".
Besides, i disagree with OP about it being a compliment to compare Anime to Spongebob squarepants, Ed Edd 'n' Eddy, Cow and Chicken (Which are easily some of the very worst/best examples of what kind of trash western cartoons are still filled with today. If u want good examples, then use Tom &Jerry, Tex Avery or Flintstones. If u want more modern cartoon examples, then use Foster's Home for Imaginary Friends, Phineas & Ferb, Secret Saturdays, and Ben 10). There are good western cartoons, but they are nowhere near as varied as Anime, aren't treated as anything more than "kid's shows" (with the notable and excellent exceptions of Simpsons, Family Guy, and Futurama), and the creators behind them don't even attempt to take their work to the levels of movies or other respected live-action media.