Conil has a classic Atlantic Andalucian climate: hot, dry summers and mild winters with occasional Atlantic rain. The water temperature in the Strait area is cooler than the Mediterranean, which keeps the beaches from becoming unbearable in July and August — but that doesn't mean the town is quiet. July and August are packed solid with Spanish families; booking accommodation weeks ahead is essential.
May and June are arguably the best months. The almadraba tuna season runs through both — bluefin tuna are caught in traditional net traps offshore, and every restaurant in town is serving atún de almadraba in some form. Temperatures are warm but not punishing, the water is swimmable, and the town has not yet reached summer saturation.
September and October are nearly as good: crowds thin out, prices drop, the sea stays warm from summer, and the light over the Costa de la Luz in the evenings is genuinely beautiful. October can bring rain.
Winter is quiet and slightly melancholy. Many businesses close or reduce hours. The Levante wind blows cold off the Atlantic. The old town is lovely when empty, and this is when locals actually use the bars and cafes.
The Semana Santa processions in Holy Week are worth seeing if you are anywhere in Andalucia at Easter. Conil's are smaller and less theatrical than Seville's, but the atmosphere in the narrow streets of the old town is intense.