It is good that you identified the 'hip' seasons as the best times to go on Camino. From June through the end of September, the Camino is more crowded, as the universities are out across Europe and mostly around the world. It is also when working folks have their holidays / vacations.
FYI, The arrivals of pilgrims at the Pilgrim Office follows a classic 'bell curve" distribution. The absolute peak is from the third week of July through mid-August. This encompasses the major religious feasts (and national holidays) of Santiago (25 July) and the Assumption of the Virgin Mary (15 August).
So, in both hip seasons, there are fewer pilgrims to compete for finite bed space. However, and as others have stated, there is more rain on the front-end (April - June). The June - September period is famously hot and drier. The weather is drier, but relatively more pleasant in September and October. But, DO remember that the rain in Spain falls mainly...on Galicia! Always be prepared.
I walk my Caminos in the April - May timeframe. This is because it is cooler (sometimes single digits in the morning), and there are lots of colorful flowers and new animals to be seen. Yes, it rains more, but that is a necessary thing to promote new growth. Conversely, in September, the crops are mostly harvested and the terrain is "browner." My choice is for cooler temperatures and more color.
Easter marks the 'official' start of the annual pilgrimage season. From Easter (Semana Santa / Holy Week) daily pilgrim arrivals at the Pilgrim Office begin an upward trajectory. What were dozens of people arriving daily in March, becomes several hundred daily by mid-April. By the end of June, the arrivals are up to 1,500 - 2,000 daily. This rate continues until mid-August. Then it starts to taper off again.
By the beginning of November, the season is essentially over. However, there are hardy folks who prefer the solitude of the Caminos during the colder, winter months.
So, for ease of finding a place to stay, walk in April - early June, or mid-September - early November. IMHO, the best weather for walking is at these same times.
Hope this helps.