In image processing, a feature descriptor is the local neighborhood "coding" of a keypoint or point of interest. Typical descriptors are SIFT, SURF, which use floating point numbers to represent the local surrounding of the point of interest. More recently, binary descriptors (which are faster to compare, using Hamming distance) were proposed, such as ORB, BRSK or A-KAZE.
A feature vector is simply a set of descriptors that represent all the keypoints in an image or a region. More generally, in classification frameworks, a feature vector is the representation for one sample of the data set in the feature space.