There are caveats. The exact strength of the privacy guarantee depends on the parameters you use and the number of computations you do, so simply saying "we use a differentially private algorithm" doesn't guarantee privacy in isolation.
I recommend watching it if you're interested at https://homepages.cwi.nl/~boncz/sigmod-pods2019.html (top-left vid)
(as a side-note Frank McSherry received SIGMOD Test Of Time Award for his Differential Privacy paper at the same conference).