Rain does not affect mail delivery unless the rain is causing flooding conditions.
If it's just raining then your mail will still be delivered but if there's flooding conditions then the USPS will have to stop mail delivery to the area that is flooding until the flooding resolves.
But normal rain does not stop or affect mail delivery.
USPS does deliver in snowy and icy conditions as long as it's safe to do so.
There have been rare cases where USPS had to stop delivery due to blizzard or very snowy or icy conditions because it was dangerous for the USPS carriers and workers to be out in the snowy and icy conditions.
If it's really bad out and the roads are really icy then the USPS may have to stop delivery of your mail for a few days but it's rare that happens.
Sometimes there may be a temporary delay for a few hours while the roads get cleared and deiced.
But USPS will make a very good attempt at delivery of your mail and packages even when it's snowing or icy.
Not everything shuts down when it's snowy or icy and the USPS will deliver in snow, rain, ice, wind etc.
I've had a few USPS deliveries not arrive for a few days when the weather was really snowy but that does not happen often.
I want the USPS workers and everyone else to be safe so I can wait for my mail and packages.