Neglect is worse than ignoring something. It's ignoring it, failing to care for it, and probably harming it in the process.
The word neglect comes from the Latin verb neglegere, which means "disregarded." You can neglect to do your chores, meaning fail to do them, but this word is usually reserved for cases when you willingly refuse to care for something appropriately. Child neglect is what parents get charged with when they fail to provide for their child's basic needs. If you go on vacation and neglect your plants, they may wither and die due to your lack of attention.