The gemara (daf 4) tells us that although bittul chameitz is sufficient min hatorah, Chazal required that one eliminate chameitz before Pesach. How does bittul fulfill the Biblical mitzva of "tashbisu se'pr m'bateicheim"? Rashi writes that since the Torah uses the term "tashbisu" instead of directly telling us to burn chameitz (tiva'aru), it implies that bittul is sufficient. Tosfos disagrees and explains that bittul works through the mechanism of hefker. The Ran elaborates: theoretically all chameitz is by definition assur b'hana'ah and ownerless, but the Torah still considers it on your possession, just like a pit (bot b'reshus harabim) in a public domain is not in your possession but you are liable for damages that occur because of it. To avoid having chameitz thrust back into one's possession because of bal yera'eh, a simple declaration suffices.
According to Rashi, tashbisu entails the active removal of chameitz from one's possession, either through bittul or destruction. According to Tosfos, all bittul does is prevent chameitz from becoming one's responsibility. It is not an active removal or destruction, but a passive avoidance of ownership reasserting itself.
The Rambam paskens that if one purchases chameitz during Pesach one would receive malkos. Any lav which is connected with a mitzvas aseh which serves as a correction to the lav is exempt from malkos - "lav hanitak l'aseh". Why should there be malkos for buying chameitz when it is subject to the corrective aseh of "tashbisu"? R' Chaim (al haRambam) answers that the aseh of tashbisu is an "issur aseh" - it does not entail actively destrying chameitz, but functions itself as a prohibitition of bringing chameitz into one's possession. This chiddush of R' Chaim may depend on the machlokes between Rashi and Tosfos with regard to the nature of tashbisu (see Minchas Chinuch).
The Sha'agas Arye discusses a possible nafka mina between the two approaches in the case of one who accepts someone else's chameitz for safekeeping, including achrayus so that if lost/stolen one would have to repay the owner. The gemara is mechadesh that even though the potential liability in case of loss is merely a "goreim l'mamom", and the one watching does not actually own the chameitz, he would violate bal yera'eh for keeping that chameitz over Pesach. According to Rashi that bittul is the same as a destructive act on the chameitz, bittul would remove the issur in this case. However, according to Tosfos, how can one do bittul on such chameitz? If bittul is merely a form of hefker, how can one be mafkir what one already does not own? It is the liability for loss, not ownership, which causes bal yera'eh in this case!