I've taken my Hennessey hammock along various lowland trails around Europe... a quality bit of kit and lightweight at 1.1 kg. In summer I don't bother carrying a sleeping mat, so I can travel even lighter.
The downsides to hammock camping: firstly you're often a fair bit more limited in where you can sleep. More frequently than with a tent I've walked much further than desired in order to find a suitable place to wild camp. With campgrounds you might well have problems too. With the right tarp it's possible to set up a ground camp, but then you'll likely need to carry a bivvy bag too (negating the weight saving somewhat).
Secondly, compared to a tent, hammocks aren't great for sitting out bad weather.
And thirdly, it depends how sound a sleeper you are, but personally I never get as good a night's sleep in a hammock as in a tent. If you can sleep on your back and don't wriggle around much they're fine, but otherwise expect to wake up fairly frequently to adjust your position. This gets better with time, but even after several months practise I still rarely feel well rested after a night in the hammock.