Every month in the Azores is a delight but the best time to go depends upon what you want to do. When scheduling your trip, there are 3 things you will need to consider: the weather, the crowds and the flower blooms.
Weather in the Azores is unpredictable but if you want to maximize your chances of having good weather, the best months are from May to September. The other months can be cold and rainy.
Temperatures in the Azores are rarely lower than 14°C, and they don't tend to get much higher than 25°C, which makes it perfectly pleasant all year round. That said, sea conditions can get rougher during the winter season so the best time to visit the Azores for boat trips (ferry travel between islands, whale watching, sailing or fishing) is definitely in the summer.
The popular times for the beaches are again the summer months and into October, but remember the Azores are not beach destinations!
For walking, the whole year is good, but the rain in winter is usually colder and heavier than in summer. In winter you will also have to take extra care in the mountains. In summer there are more eating places open and they keep longer daytime opening hours.
However, be aware of the tourist crowds in the summer months. July and August are the peak tourist season. Hotels throughout the islands are usually fully booked and more expensive. The same goes for car rentals. Car hires increase their prices during the summer months as cars are in high demand. Make sure you book your car rental and hotel well in advance to avoid disappointment..
Finally, if you are into flowers, there will always be something of interest but for the beautiful and spectacular hedgerows you should consider June and July to see them in their full glory. When August arrives they will start to get brown from sun burns but you will be able to see other native species like the Azaleas and the Yellow Lilies until September.
The best time to visit the Azores is quite simply whenever you get the chance but if you have to choose two months, I would probably recommend June or September.