For small screens Input component takes about 1/4 screen width which could be quite annoying.

The same method of hiding a component as in Legend could be applied here.

Maybe it would be useful to allow user to enable/disable the geojson layer.

Large json data can be saved on the user's browsers using Indexeddb. I have to serve several big files (the largest is 7.5MB) to make it all work without any backend.
It could be good idea to save those files using one of Indexeddb wrappers (localForage or react-indexed-db).

After first clicking on a layer, it takes some time to load all necessary data.
Splitting those weather data files based on the month would improve the loading time of the first popup. It would also mean that the user would have to initially wait for the data to load each time he/she changes a month.

Changing all layers on the map (when the month or the data type changes) is quite RAM and CPU demanding. Some optimization is needed here, especially in the Map component.

When loading PageInfo component sometimes it takes additional 0.5-1s to show React Leaflet logo. It'd be much better to show the whole page after all dependent imgs and fonts are loaded.

Some places in the geojson file don't have any name assigned or are not assigned to the proper admin1 division.

Currently, the Input component uses window resize event to set isSmallScreen state. I'd be much better to use something less resource hungry like react-resize-detector.