Bloomfield came from the Neogrammarian school of linguistics. That means he focused on the historical aspects and evolution of languages. He studied particular languages, their history and how words are generated. Both Bloomfield and Saussure studied language as a structure or with a scientific basis. The main difference is that Bloomfield studied linguistics diachronically: it's historical and comparative development. Saussure studied language synchronically: he made the comparison between language and chess. You don't need to know the history/previous moves; you could understand the system just by looking at the board at any single moment/state. This is the synchronic. Unlike Bloomfield who studied the historical development of linguistics, focusing on behavior, Saussure sought to understand the underlying structure of language in general: and this had to do with the relationship between signs. Bloomfield's study would focus more on this history of signs. I think you could say that Saussure's study is more abstract (still scientific) and philosophical and Bloomfield's was more sociological and a little psychological.
Bloomfield and de Saussure are both Formalists, they studied the structure of the language. De Saussure was the founding father of modern linguistics. Bloomfield was used to study language diachronically but he converted to the tendency of the 1920s of B.F.Skinner Behaviorism. Bloomfield sees language as a set of verbal behavior. Natural languages are triggered for physical need. Language is acquired through pairing repeated Stimulation (stimuls -->response) ( you can look on the example of Pavlov) Bloomfield was known for his scienticism, behaviorism and positivism. DE Saususre rejected the positiivistic view of the word and its objects. de Saussure emphasized that words are consulted in terms of signification i.e. signifier --> signified which resulted his theory of Semiology.