If you said 'while it hasn't been raining, the street is wet' it works. the other way round does sound a bit strange. Sadly, I am going to hide behing the native speaker's defence of 'it sounds' rather than coming up with a sound reason.;-(
Let my ask one more thing. If the sentence were like "The street is wet, while it is not rainy", is it possible to take "while" as "although"?
The weather is rainy. It is raining.
For some reason "while" does not fit. Try the common sense approach - the street is wet - no rain. You expect it to be wet when it rains. So - why is it wet? I think that is where "although" comes in place.
In September the florist carries tulips although they are not in season.