Shulchan Aruch O.C. 448:5 rules that chometz which is found in your home after Pesach, even if you nullified it, is forbidden. The Tur there (cited by Magen Avraham and Taz) explains that we are worried that a person will say he nullified it even if he didn't. This isn't a problem for those crumbs that you don't want anyways. But for that box of cheerios you were planning on enjoying, it's a problem.
There is another reason that could be used to explain why it would be forbidden to use your chometz after you nullified it. If you nullify your chometz and declare that you consider it to be like dirt, and then start munching on it 9 days later, it would be a little bit of a contradiction to your nullification of it. (I can't remember where I saw this idea explicitly, but it is similar to the idea (Taz O.C. 442:8) that if you eat chometz which is degraded enough to not require getting rid of it, אחשביה, you have given it importance and reinstated its negated status.)
If you sell your chometz, you don't have these issues.
Selling your chometz would certainly not be prevented by having done bittul according to many Rishonim. While Tosefos (Pesachim 4b s.v. mid'oraisa) seems to hold that bittul accomplishes hefker, the implication of Rashi (ibid, s.v. b'bittul b'alma) and the explicit claim of the Ramban (Novelae to Pesachim 4b) is that bittul does not effect hefker and you still own your chometz. As the Ramban explains it, with bittul you are declaring acquiescense to the Torah's prohibition of chometz and submitting to the Torah's removal of chometz from your ownership. This happens after your sale takes effect. According to this view, you can "nullify" your chometz and still have ownership until the Torah removes it from your ownership at the 6th (or, Rabbinically, 5th) hour.
You would still want to do bittul, either to fulfill the mitzvah of tashbisu according to those who hold it is a fulfillment of the mitzvah (Ramban novelae to Pesachim, among others), or just to fulfill the Rabbinical enactment.