I'm intrigued by the idea of hammock camping. Most of what I have found out is as
@Heltor Chasca says above - difficulty in finding suitable locations and insulation issues leading to bigger/heavier things to keep ypu warm. I also have the idea that while a hammock sounds like a great idea, the times I have tried a (regular) hammock in real life the reality doesn't live up to the anticipation.
The tarp idea is a different kettle of fish. At its simplest a tarp and a pole (or more likely for hikers, walking poles). Simple, light but lacking comforts such as an inner to keeep bugs out. Of course, you can buy an inner if you like. "Pure" tarp tents allow you to make different designs according to your needs on a given pitch, which is very handy, especially for hikers. Let's face it, if you're on a bike you can always ride on looking for somewhere more suitable.
I've always had the idea that tarps and the like were for the real bushcraft enthusiasts, those who get a kick out of making the best camp possible. That might be different for a cycle tourist who just wants to get their head down in reasonable comfort.
I think, also, that their use requires a bit more knowledge/skills/experience to get the best out of them as opposed to standard, big brand tents.
Last year I got a Hexpeak4A tipi shelter with a one person inner. It's very light, so far strong enough, decent space inside and very usable (for 1). Of course, it needs to be fully pegged out, but there is a degree of flexibility in terms of the height I set it at to increase or decrease ventilation and the inner can be easily dropped to give me the full use of space. The inner is a bit saggy, something I might improve, but that will require a little DIY. There are numerous options on Ali-Express for different inners, some of which get good reviews.
I used it on a 3 week tour last year and it was perfect. I'm not sure if I'd use it for a long term tour, the least attractive point is that it is not freestanding.
On that basis, I'd think a tarp tent would be something of an ongoing project, making improvements as you go along. You'll see users in the States with fireboxes in their tents, for example.
I'm not sure, but most of these style tents that I have seen for 2, have a double inner which changes the set-up to inner first and reduces the usable space between the inner and the fly. Something to consider if there's 2 of you.