Lightroom Classic: Local catalog, images stored locally on hard drive, internal or external, if needed possible to have multiple catalogs, can send smart previews to the cloud. Basic Lightroom Classic works similar to Lightroom that has been around for several years.
Lightroom: Images are stored in the cloud. Possible to store a copy locally. Images are available to edit on all devices. No multiple catalogs. Feature set more limited than Lightroom Classic, but not critical for those who like Lightroom. The Lightroom plan includes 1 TB of cloud storage.
These are very basic differences. Lightroom, among other things does not have a print module and several other features that are available in Lightroom Classic such as the book module.