Both periods are good. I went to the Norte in early October last year. There was an unusual heat wave in Spain, but areas close to the coast are always a bit cooler. The weather was very fine, with occasional light rains.
Based on my experience on the Frances (not the Norte), there could still be more serious rains in late April, which can lead to muddy paths. It is nothing problematic, but you need to be prepared for it. I personally like more this season.
Since (I suppose) you are planning to walk for a month, you can expect some weather changes.
Regarding the crowds, it is inevitably a trade-off. Good weather inevitably means more people. Your last two stages, where the Norte connect with the Frances, is always crowded, but is also quite fun.