Differences Between Protein Motifs and Protein Domains?
These are completely different concepts, which sometimes may be connected.

A motif in biology is a mathematical model, typically of a sequence, that predicts which sequences belong to some defined group. For example, a DNA sequence motif can characterize the binding site of a transcription factor, i.e. which sequences tend to be bound by this factor. For proteins, sequence motifs can characterize which proteins (protein sequences) belong to a given protein family. A simple motif could be, for example, some pattern which is strictly shared by all members of the group, e.g. WTRXEKXXY (where X stands for any amino acid). There are also more complex motif models.

Protein domains, on the other hand, are a structural entity, usually meaning a part of the protein structure which folds and functions independently. So, proteins are often constructed from different combinations of these domains.

So how are motifs and domains related? Well, when you think about protein families, it makes sense not only to look at the whole sequence but also to focus on individual domains. Since they are a elementary functional-structural units, it makes sense to find sequence motifs for individual domains. So, you often find that a protein contains multiple domains, each domain characterized by having a sequence that matches the motif of its family.