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The GRIB file format is designed for storing and distributing weather data. GRIB files are is widely used in meteorological applications (Wikipedia article, including a list of software to read GRIB files). GRIB files usually have the extension .grib, .grb or .gb.
For software developers and users comfortable with command line tools we recommend GRIB API, developed at ECMWF to decode and encode GRIB data. For geospatial applications you can also use the GDAL library.
The NetCDF file format is a more generic format, commonly used to store and distribute scientific data (Wikipedia article). NetCDF files usually have the extension .nc.
For example, ERA Interim, the main dataset in the Copernicus Climate Change Service (C3S), is produced by ECMWF in the GRIB edition 2 format. The data is then made available for download in its native GRIB format, and also in a version converted from GRIB to NetCDF. Note that due to limitations of the NetCDF specifications the NetCDF version contains fewer parameters (variables) and incomplete metadata. The NetCDF format is not formally supported by ECMWF.
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