Core promoter refers to specific sequences which RNA polymerase binds to. It is the minimum stretch of DNA sequence which is enough to direct the accurate initiation of transcription by the RNA polymerase.
Promoter proximal regulatory elements refers to sequences which allow eukaryotic cells to express certain genes only while repressing others. It generally refers to any regulatory sequence in eukaryotic DNA that is located close to a promoter and which binds to a specific protein, thereby regulating the transcription of the associated protein.