To add MODIS Basemap layer and combine it with current weather layers, you need to specify the following endpoint URL's to both of the tileLayer as it for Leaflet-js.

http://{s}.sat.owm.io/sql/{z}/{x}/{y}?select=b8,b12&where=day>2016-01-01,day<2016-10-30&from=s2&op=ndvi&APPID={APIKEY}
http://{s}.tile.openweathermap.org/map/precipitation/{z}/{x}/{y}.png?APPID={APIKEY}

See the interactive example at the Basemap description page.