Rileys Freaky Fridays

I looked this up on dictionary .com , Because like you I didn’t know thi difference either, Hope this helps.

bi·o·chem·is·try
1. the science dealing with the chemistry of living matter.
2. the chemistry of living matter.

chem·is·try
1. the science that deals with the composition and properties of substances and various elementary forms of matter. Compare element (def. 2).
2. chemical properties, reactions, phenomena, etc.: the chemistry of carbon.
3. the interaction of one personality with another: The chemistry between him and his boss was all wrong.
4. sympathetic understanding; rapport: the astonishing chemistry between the actors.
5. any or all of the elements that make up something: the chemistry of love.

Biochem is like a combination of biology and chem (duh)

Unless you are going to do something to do with living things, just do chem.

Biochemistry is the branch of chemistry that deals with the chemistry of living organisms.
Chemistry is the wider field that also includes inorganic chemistry, organic chemistry and physical chemistry.

Studying chemistry is more useful if you are looking for a career in chemical engineering or materials science. Biochemistry is more useful if you want to go into the biological sciences or medicine. I don’t know what synthetic biology is, but I would guess from the name that biochemistry would be the more useful field of study.