Rain is water falling from the clouds, while a rein is the strap used to steer a horse, and by extension, to rein in something means to hold it back. If you're picturing weather, it's rain; if you're picturing reins in a rider's hands, it's rein.
The core difference
Rain is weather; rein is about a horse's strap, or restraint.
- rain — water falling from clouds, or to fall heavily: the forecast warned of rain across the region all weekend.
- rein — a strap used to guide a horse, or to restrain: she kept a tight rein on the household budget.
How to tell them apart
The confusion usually crops up in the phrase free rein, which people often misspell as "free reign" or even "free rain". The correct phrase is free rein, because it comes from loosening a horse's reins to let it move freely, not from weather and not from ruling as a monarch.
A simple check: if you can replace the word with control or restrain, you want rein. If you're talking about the weather, it's rain, no exceptions.