These two figures are based on two different surveys that use different definitions. The new jobs number comes from the establishment (payroll) survey. The unemployment number comes from the household survey. It says this right in the BLS employment report- it is literally in the second paragraph in the report!
For comparison purposes, BLS produces an adjusted version of the household survey that matches the definition of the payroll survey. See the chart. The adjusted household survey saw a gain of 394k jobs in May, which is actually higher than the payroll survey's 339k.
While they tend to trend with each other divergences can happen. There is sampling error between the two surveys. The household survey is a smaller sample than the payroll survey, so tends to have greater error.
There are always so many dumb conspiracy theories about the employment and unemployment figures.