If you asked whether a whole phrase or just another word is a synonym, would say that yes.

Why are two words synonymous? Why do we make distinctions between “Exact Synonyms” and “Nearly Synonyms”? I’d say that these two phrases are basically synonymous. What are the examples? When search and look are used in this precise context, they carry very similar meanings. Simplifying terms (not in different sentences) is impossible. On the other hand, search carries out the connotation of a greater or more exhaustive process. Look could be much more casual. This connotative distinction follows naturally from the distinction between the independent meanings of looking (active seeing) and searching (actively and methodically looking with intent to find something) outside the context of this phrase.