Earlier in the season, Jane told Jessie that they need to fall asleep on their sides after doing heroin. The reason is if their body had some kind of violent reaction to the drug, causing them to vomit, they would asphyxiate on their own vomit. However, if they were laying on their side, the chances of the vomit "pooling" in their mouth were reduced, thus the chances of asphyxiation were reduced.
Jane turned over (or Walt turned her over, or she didn't fall asleep on her side, I can't quite remember the exact details), had a bad reaction to the drug, and began vomiting, but since she was unconscious or unable to move due to the drug, she could only lay there and asphyxiate on her own vomit, unaided by Walt.
The same thing didn't happen to Jessie because he didn't have a bad reaction to the heroin they injected, thus there was no vomit to suffocate on.