They solve two different problems
An applicant tracking system manages people you hire directly: sourcing, screening, interviews, offers. A vendor management system manages the suppliers who send you contractors: rate cards, requisitions, timesheets, compliance, spend.
Companies get this wrong in a predictable way. They buy an ATS, then start using contractors, then run the contingent workforce through spreadsheets because the ATS was never designed to track a rate card or an insurance certificate expiry.
Choose by workforce shape, not headcount
If almost everyone you bring on is a permanent employee, an ATS is sufficient and a VMS is overhead. If a meaningful share of your delivery capacity comes through agencies or contract suppliers, the VMS is the system that stops that spend from being invisible.
The threshold in practice is lower than people expect. Once you are working with more than three or four suppliers, manual vendor management costs more in coordination and compliance risk than the software does.
Running both
Most companies past a certain size need both, and the value is in the join: total hiring cost and time-to-productivity across direct and contingent hires, in one report. That is the argument for taking them from one vendor rather than integrating two.