There is nothing wrong with the first sentence. Its fairly standard English (at least I assume since I understood it with not a second thought). The only thing that might have thrown the poster above off is the formality of it. i.e. while grammatically correct, 9/10 you won't hear:
Thursday will be colder, but we will have less rain.
Instead you'll hear:
Thursday will be colder (than x-day), but we'll probably have less rain (than x-day).
The flow was off. It was missing something to be compared to. You can put the day being compared to in either independent clause, but not both unless the days are different. E.g.
Thursday will be colder than Wednesday, but we'll probably have less rain than Monday.
That sounds perfectly natural.
And you probably need a certain expression of doubtfulness so I threw the probably in there. Usually, only weathermen (or the people who've watched them) are so certain about the weather as your OP implies.
To the ACTUAL QUESTION:
I don't think its that hard. Don't think of But with as one expression; rather, think of it as 2 expressions with fused meanings.
A but with B = A but A and B
In other words, I think your first sentence and your second sentence have little relation to each other aside from the fact that they mean the same thing.