Voted bivy.

I have used both a lot. Hiking - perhaps 50+ nights in hammocks, and 500+ nights in a bivy. Biking - 4 nights in a hammock and perhaps 50+ in bivy.
However, while bivying, less than a dozen nights were spent in a bivy alone, the rest of them I had a (shaped) tarp pitched above. The pure bivy nights were all in alpine areas, in winter or shoulder season, all of them during mountaneering and alpinism.

The hammock is great for very rough terrain as I don't care what the topology is under my ar$e when I hang. However, during bike toruing such terrain is almost non-existent near roads and trails suitable for biking.

The hammock has 2 drawbacks, which have completely eliminated hammocking from my bike tours:
1. Weight. My bivy sleep system is always lighter than the lightest hammock system, and I have tried numerous and still own many.
2. Bulk. In a hiking backpack this is not that much of a detriment, but on my bike it is a huge one. I tour light and with very limited luggage volume. Hammocks don't fit this picture.
3. (I know I said 2 drawbacks...) Needs provisions to hang it (trees, poles, large boulders, improvised scaffolding - I've done all of these), but this one is relatively minor unless you are travelling through the prairie, and definitely minor compared to No. 2.

All of my bivys have nets to protect me from bugs during sleep, and close well with zippers. Under heavy bug pressure I sometimes put a headnet in camp while cooking or milling about before I get into the bivy to sleep, although I don't spend much of my time this way. When I arrive at camp, I wash if I could, cook sometimes, eat sometimes (if not eaten recently somewhere else prior to stopping for the day), and go to bed.

The tarp overhead protects against rainy, very windy, stormy weather, and ensures great protection against cold in particularly cold nights under clear skies, when there's not a lot of vegetation above me, as it drastically reduces my body's heat loss through infrared radiation. It also allows me to use very light and really breathable bivys, which is super important in the areas I travel most - Appalachians - where the high humidity and dew point drives lots of moisture issues in less breathable bivys.