The stellar-base library is the lowest-level stellar helper library. It consists of classes to read, write, hash, and sign the xdr structures that are used in stellard. It doesn't assume you are talking to horizon.
stellar-core is the backbone of the Stellar network. It maintains a local copy of the ledger, communicating and staying in sync with other instances of stellar-core on the network. Optionally, stellar-core can store historical records of the ledger and participate in consensus.