Quick answer
April–June and September–October are the sweet spots: 21–25 °C, low rainfall, sea around 21–22 °C, jacarandas (May) and vineyards (Sept). July–August are warmest but busiest. November–February are cheapest, greenest and best for hiking but the sea is too cold for most visitors.
Context
Madeira's climate is subtropical, so 'best' depends on what you came for. For all-round weather plus reasonable crowds, late spring (mid-April to mid-June) and early autumn (September to mid-October) win on almost every axis: warm but not hot, dry but not parched, sea warm enough for most swimmers, hotels still 20–30 % cheaper than peak.
If you've come to hike, the laurisilva is at its lushest in winter and the high peaks are clearest after a cold front in January or February. If you've come to swim and sunbathe, July and August deliver the most reliable beach weather but you'll share Calheta and Machico with everyone.
Avoid the last week of December if you want quiet, Funchal's New Year fireworks pull a Madeiran-diaspora crowd that fills every hotel.