I'm not a native speaker, but I would translate them as follows;
1 It was raining yesterday, but it's stopped today (it's not raining any more)
2 It rained yesterday, but it hasn't rained today.
In first sentence the weather seems to be a more continuous event that goes from one day to the next, where as in the second the weather on the two days seem to be more separate events.
The phrase 不。。。。了 ， means not....anymore.
不下雨了 - it's not raining anymore
我不学中文了 - I'm not studying chinese anymore
As said I'm not a native speaker, so would be a good idea to get someone to confirm this!