I was there in Year 2000. Kilimanjaro is much higher than Rainier, so it might be colder. OTOH, it is much closer to the Equator, so it might be warmer. It all depends on which way the wind blows.
I had taken a medium down jacket with me, intending to use it around the summit. Then somebody else got sick and cold at 18,800', so I loaned my down jacket to her and went up without it. As long as I kept several thin warm layers on with my goretex parka shell over them, I was OK.
Part of the problem is that you are trudging upward for hours and hours at a time, so you get your body heat ventilated and managed. Then suddenly you are descending for hours and hours at a time and you get colder. Whatever you wear, keep it adjustable. Some people get exhausted from the high altitude, so they start sitting down a lot, and that tends to get bad results. It's probably better to keep moving upward, even if it is very slow.
If I had it to do again, I might substitute a light down jacket for the medium one.