Enhance has more to do with increasing quality: We used Photoshop to enhance the image. The updated computer has enhanced features. Boost has more to do with increasing the amount or height: We boost sales with our advertising campaign. Someone can boost someone else up (lift) so that the person can climb over the wall.

'Boost' is usually considered more dramatic. We talk about a booster rocket that adds a lot of extra speed to a missile. We don't talk about a rocket that enhances the speed.