Ilulissat divides neatly into two seasons and both are genuinely worth visiting.
Summer (June to August) brings the midnight sun — from about late May to late July the sun never sets, which means you can hike to the icefjord at 2am and have full daylight. Boat trips among the icebergs run daily, whale watching is excellent from July onward, and the landscape is green and surprisingly warm by Arctic standards (10–15°C on good days). July is the peak tourist month.
Winter (November to April) is dog sledding season. The sea ice freezes, the northern lights appear (September through April, with peak activity around the equinoxes), and the town takes on a completely different character — quieter, more local, and genuinely dramatic. Temperatures drop to minus 20°C and below; dress for it.
March and April offer the best of both worlds: the days are lengthening rapidly, temperatures are still cold enough for good sledding, and the northern lights are still visible on clear nights.
Avoid May and October if possible — the ice is neither stable enough for sledding nor absent enough for good boat access, and the weather is raw.