Q.What is the difference between water tight doors and weather tight doors?

Water tight door - They are designed to take the pressure of the head of water they are designed for. If ballast tank has a door at the lower end and water fills upto 15 mtrs in tank then  that door should be able to hold that water pressure, infact from from any side. Any compartment having water tight door then water should not leak out or enter in.

Wheather tight door -- They should not allow water to enter in the compartment in any weather condition so means from from outisde (one direction). Like hatch covers will not allow water to go in the cargo hold from outside in any weather condition. Or like accomodation weather tight doors. But weather tight door will not take the pressure of the water. That means if in above condition we had weather tight door then at 15 mtrs it will leak or give away.

Main to remember is Water tight - both sides and can take presure head. Weather tight one side and not design to take pressure of water.

Easy thumb rule- All opening above waterline are weather tight. All openings below waterline are watertight. 

You can find explanation in Solas.i think chapter 2 part A.