I'm a weather nerd and have been so since childhood, when I discovered a Golden Nature Guide about weather in our family's home. This is going to be a very brief rundown of the apps I currently have. I have a Android phone, but I believe all three are also available for iPhone.
Windy
Windy is a sort of Swiss Army knife, and it has so many features that there are some I've never learned to use. The app opens to a map showing the winds blowing over a large area, expressed as animated arrows showing the direction of the wind and (by the thickness and length of the arrows) its force. The map can be zoomed by pinching or spreading, and panned by dragging. Coverage is available wolrdwide. A hamburger menu in the lower right gives access to a number of different views for the area shown on the map: weather radar, satellite, rain/thunder, temperature, and more. An interactive bar on the bottom of the map shows date and time; you can slide the bar to display past conditions or forecasts. Windy also has a website with many of the same features, if you want to check them out before downloading the app.
MyRadar
I got this one because Windy's radar map didn't give the level of storm detail I wanted for winter snowstorms or summer thunderstorms (weather in the Washington, DC, area is notoriously hard to predict at the county by county level, and even within our county, there can be crucial differences between the north and the south). MyRadar is good for what it does.
Today Weather
There are lots of general weather forecast apps out there. I wanted a functional on-screen widget, specific local forecasting, and a minimum of ads. Today Weather delivers. The widget is customizable, and the internal display shows your current location's temperature, UV index, etc. in a summary block, followed by a week of brief day-by-day predictions, an hourly precipitation forecast for the next 24 hours, AQI, pollen counts, sunrise/sunset, moon phases, wind, and radar. I usually see only a single inline ad after I bring up the app.
The one thing that's mildly buggy is that the widget takes a minute or two to reappear after you've restarted your phone.
I should note that in the case of a fast-moving weather situation near to home, I still refer to the Washington Post's Capital Weather Gang feature. As disappointing as the WaPo's recent editorial changes have been, it's still my hometown paper and it still has the best weather coverage for the DC Metro area.