Is airtight tighter than watertight? If something is airtight does that mean it's also watertight?
If something is watertight do I need to worry that it might not be airtight, or can I assume that it is?
So which is tighter and does 1 include the other?
 A: Most things that are "airtight" or "watertight" actually aren't--that is, they permit some small flux of molecules. So the answer comes down to practical considerations--if the leakage is slow enough not to matter for the intended use, then you can say it "doesn't leak".  Most people would say that car tires are "air tight", but they go flat eventually, as air molecules diffuse through the rubber or find their way through microleaks. In some situations, even much slower leaks than this are large enough to matter--in processes involving extremely high vacuum, rubber o-rings can't be used for sealing, because gasses diffuse through them. 
That said, water vapor is a constituent of air, so something that is truly 100% airtight is also watertight: it does not allow water molecules to pass. As has already been pointed out in comments, this is not necessarily true in the other direction: there are constituents of air that are smaller than water molecules, so presumably there is a hole size that will permit some air but no water.
A: One does not imply the other.
Most molecules that make up the air are smaller than water molecules (e.g. $Ar$ is smaller than $H_2O$), so it's easier for air molecules to diffuse into the container. 
However, on the other hand, there is more pressure under water, so the container that is only designed to be air-tight may break if the pressure is high enough.
