The enzyme used by CRISPR is Cas9 (which is why you'll often see it referred to as CRISPR-Cas9), which acts like a pair of molecular scissors and cuts strands of DNA at a specific point in the genome.
Base editing is a newer technique and utilizes an enzyme called deaminase, which can change the chemical structure of a base pair without doing the "cutting" part. There are specific deaminases based you use depending on which base pair you're trying to modify.
To oversimplify a bit: base editing is a more precise version of CRISPR, which is less likely to cause more changes than actually desired, as is a problem with CRISPR. That doesn't make CRISPR obsolete though, because if your goals actually is to remove entire portions of a genome, CRISPR is a more effective technique. Regardless, there is a lot of overlap between the methods: both usually use adenoviruses as the delivery vehicle, you can use both enzymes in tandem, etc.