I'm on holiday at the moment and didn't bring my dictionary (duh) but you may still find the following useful. Complacent ~ Satisfied almost to the point of carelessness and not looking to improve skills. For example: The aircraft pilot could not allow himself to be complacent when landing on Luton Airport's short runway. Conceited ~ Proud and believing that you are 'better' than others. She was a very conceited woman who felt it was beneath her to speak to the other women in the room.

Hello, Complacent : self satisfied, unconcerned specially when accompanied with unawareness of actual dangers or deficiencies . It is mostly a negative trait ,almost a careless behavior . Conceited : having or showing an excessively high opinion of oneself .

I think both are very similar; both can be used negatively.