When I use smooth vertices operation, vertices does not stay snapped on the surface.
I have to select single lines and snap it again on the surface.
Does anyone know why it happens?
Smoothing is indeed not affected by Snapping, only Move (and Rotate and Scale, if enabled).
I always do Smooth Vertices, G, Enter. You don't need to select single lines, you can Snap a whole area.
You can also, use a Shrinkwrap Modifier, in addition to Snapping. (Enable the On Cage 'triangle icon', to see it in Edit Mode.)