I think every time you go, you will see something different. So it is highly dependent on where your interests lie.
Summer ( mid-June to end of August)
There is no darkness in the summer; the sun sets for a couple of hours, allowing for fantastic pictures opportunities. You can travel during the night and be alone everywhere during that time. On the downside, it is hard to sleep when it is continuously bright, and you lose the notion of time quite fast:) Summer is also a time where the most tourists visit, so attractions on the Golden Circle route are quite crowded.
Spring and Fall (Mid-May through mid-June, September through mid-October)
During those time there are much fewer tourists around, meaning the prices for accommodation drop significantly.
The later in the Fall and earlier in Spring you come, the bigger the chances to see northern lights are.
Winter and beginning of Spring (Mid-October through mid-May)
Many attractions become accessible during this time, for example, glacier caves or northern lights. Majority of the main roads are fine, but some of the passages in Highlands close due to snow. The problem is long nights; in December there are around 2 hours of daylight. I was told it is cozy time though, as there are a lot of Christmas lights all around. Winters are also milder than what you expect, and Iceland is warmer than New York during this time.
Time-specific activities:
Whale watching
The best time of year to see whales around Iceland is from April to October with the peak season in June, July, and August. Time doesn’t matter, most companies claim they have 98% spotting chance, and if you don’t manage to see one, you try your luck again for free. There are several places you can do that from, so it all depends on your itinerary.
Puffins
Best time to spot puffins is at the end of April until early September. Best time of the day to see them is in the morning (07:00-10:00) and the evening (18:00-22:00).
Northern Lights
The Northern Lights season is from late August to mid-April. However, from late September to late March, it is dark after 6 pm, so the chances grow the darker it is.
Highlands
For most places in Highlands, you need 4×4. Many minor roads are closed from late September to June (and sometime early July) due to snow and ice.
Glacier Caves
The only time to visit the ice caves is in the winter from mid-November until about mid-March. For the rest of the year, the temperature is warmer, causing the glacier to melt slowly. It might cause the cave to collapse; that is why it is too dangerous to let tourists in during summer.
