Differences are mainly in corrosion resistance. 308 is generally used as a wire for welding 304. 316 contains some molybdenum and has better corrosion resistance in many environments, sometimes referred to as 'marine grade' as it's less prone to pitting and crevice corrosion in chloride environments than 304. 317 is more highly alloyed still for harsher environments
There's also L grades (low carbon), H grades (higher carbon) and others like 316Ti that effect properties, particularly at higher temperatures. Adding carbon increases strength but at higher temps the carbon can nick some of the chromium (carbide precipitaion) lowering corrosion resistance. The L grades are immune or it can be prevented by adding some titanium, Ti has a higher affinity for C than Cr does so the carbon nicks the Ti instead leaving the chromium available to do it's oxide film thing
All very generalised, google will get you much more detailed info. Tons and tons on 304/316, 317 is a bit rarer so try googling '317 stainless steel'
