For me, the big thing about hammock camping is comfort. I backpack the AT in Virginia alot and the rocky soil is something my back no longer tolerates even at my young age. Nothing worse than waking up the next day with a sore back and shoulders and having to hike 12 miles over to the other side of the mountain.
Plus I always hated having to look for preexisting campsites to find semi-flat ground to set up my tent. Now I can set up my hammock wherever I need to. So location choices is also a big advantage to hammocks.
As far as winter camping in sub-freezing temps with ample snow, watch Shug's Videos. Hes the expert on that.
In the Summer Time, hammocks own the forest. Much lighter weight than a tent and waaaay more comfortable. Also, much cooler than a tent. Plus during downpours its nice to sit in a hammock an look out around you vs being locked down inside a tent staring at 4 walls.
Just my personal likes about hammocks vs dislikes about tents.