While both BPSS clearance and BS7858 vetting are employee screening standards, they serve different purposes and apply in different contexts.
BPSS (Baseline Personnel Security Standard) is a basic pre-employment screening standard used for employees and contractors working in government agencies, defence the civil service or suppliers tot he public sector. Its also the precursor to more advanced government security clearance.
BS7858 is a more comprehensive security screening standard used for security sensitive roles, such as in the security industry, healthcare and education. BS7858 vetting covers the same checks as BPSS clearance but with a more detailed background investigation, with additional screening to cover financial probity and a five years or ten year employment history depending on the nature of the role.