I use a precipitation radar app that tells me with fair amount of accuracy whether/how heavily it's going to rain 15-90 minutes from now. It often does matter what part of the city you're on. Since we bike everywhere to get around, it's kind of nice to get an idea whether waiting another 10 minutes to leave avoids the heaviest part of a shower, or better just bite the bullet because it's only getting worse in the next hour.

Although now that I think of it, I have its location locked to my home address because I didn't feel like being tracked :) It generally works well enough within ~3km or so, and I compensate the hit in accuracy by looking at the sky and drops/splashes in puddles/windows/cars (it's much harder to judge the intensity of rain by trying to spot droplets in mid-air).

I don't need the longer term forecast quite as much to have an app for it, for that I just use a bookmark to my local news weather page.

Not to the meter but we get plenty of rainstorms where a half mile makes the difference between wet and dry, and Dark Sky’s push alerts are surprisingly useful.

There isn't weather stations every half mile.

They use radar and software to predict storm growth and trajectories. Next time you look at a radar map ask yourself the physical size represented by one pixel: in much of the United States it should be something like 150m.

Irrelevant, radar has accuracy at that level.