The easiest way to distinguish these two tests, is by looking at the reports.
In general, a penetration test is more focused, more direct, and more detailed. A vulnerability assessment is designed to enumerate the threats to your internal critical resources; penetration testing, alternatively, is designed to exploit vulnerabilities as a proof-of-concept.
Penetration tests are intended for organizations that have developed strong information security practices over time and are ready to put their efforts to the test.
Vulnerability assessments are intended for organizations of all sizes and maturity levels and tend to be utilized more often.