The short answer: SAP is the assessment; the EPC is an output
A SAP calculation models the predicted energy performance of a dwelling and is used to demonstrate compliance for a new home or certain conversion and extension projects. An EPC is the certificate that records a building's energy rating. For a newly completed dwelling, the final EPC is generated from the as-built SAP assessment.
New build or residential conversion
A new dwelling normally needs a design-stage SAP calculation before construction details are fixed. At completion, the model is updated using the final specification, air-test result, commissioning information and evidence of installed products. The as-built SAP reports and on-construction EPC can then be produced.
Extension to an existing home
An extension does not normally receive its own EPC. Compliance may be demonstrated through elemental standards or, where glazing or design choices go beyond the usual limits, a SAP-based whole-dwelling calculation. Building Control should confirm the route required for the specific project.
Selling or letting an existing home
An existing dwelling usually needs a domestic EPC survey rather than a full design-stage SAP assessment. The assessor records the property, construction, heating, glazing and evidence of insulation using the applicable existing-dwelling methodology. If the property is newly created or not yet completed, the on-construction route may apply instead.