If you look at the Google ngram for the term ‘Molecular Biology’ you will see that it first appears just before 1960, and it is relevant that the Journal of Molecular Biology was founded in 1959. It is not biochemistry — indeed the name was meant to indicate its difference from biochemistry, and I heard second-hand (so I cannot vouch for its truth) that Hans Krebs (Professor of Biochemistry at Oxford University at the time) referred to it as “biochemistry without a license”. What I can vouch for is that the contempt was mutual.
At the outset the term ‘molecular biology’ was used to combine two rather disparate areas of biological science that today we might refer to as Structural Biology and Molecular Genetics. The link between these was the work on the structure of DNA and proteins at this time, and the molecular analysis of the genes that controlled the synthesis of proteins.
If you look at the definition in the short Wikipedia article, only Astbury — who worked on protein structure in Leeds — is quoted, but The Journal of Molecular Biology was initially almost the house journal of MRC Cambridge, the love-child of Crick and Perutz, as it were, with an emphasis on nucleic acid as well as protein structure, leading to the other side of the coin which was the genetic code, the lac operon and the study of bacteriophage.
Of course, terms change their meaning as science changes and what was once new and trendy becomes “yesterday’s girl”. It really doesn’t matter that much. Systems Biology anyone?