The best time to visit Puri is October to February, when temperatures are manageable — warm rather than punishing — and the Bay of Bengal is calm enough for swimming. December and January are peak season for Indian domestic tourists, particularly over Christmas and New Year, so expect the beach hotels to be busy and prices to reflect it; book ahead. November is often the sweet spot: post-monsoon freshness, before the holiday rush, and the sea at its most inviting.
March to May turns hot and progressively more humid as the heat builds toward the pre-monsoon peak. April and May can be genuinely brutal, with temperatures above 40°C and humidity that makes the heat feel worse. Not impossible to visit, but not pleasant unless you have specific reason to be here.
The monsoon arrives in June and runs through September, bringing heavy and sometimes relentless rain. The sea becomes rough and swimming inadvisable. And yet: June or July is when Rath Yatra happens, and if witnessing one of the largest religious processions on earth matters to you, then the heat and the rain are a price worth paying. The city's transformation during Rath Yatra — the millions of pilgrims, the enormous chariots, the scale of collective devotion — is something that cannot be experienced any other way and at any other time. Go for the festival, accept the weather.