Moving rope (DdRT):
+ some mechanical advantage that helps returning from limb walks, short ascents, etc.
+ rope actually only sees half your weight
+ more rope in the system, between you and the anchor point, shock loading mitigation
+ plenty of effective devices/techniques for ascent/descent with long, time proven history
+ The ZigZag! The Spiderjack! The Hitchclimber System!
+ easy canopy anchor placement, install and retrieve from ground
- although effort is reduced, travel up the rope is halved... each foot of rope travel through the ascender only gives you half a foot of ascent
- doubled rope can be harder to manage in the tree.. bulkier, heavier
- redirects usually involve using a double-sheave pulley, especially if there is more than one redirect, due to large amount of friction from the moving, doubled rope
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Stationary/Single rope (SRT):
+ rope travel is 1:1 ...a foot of rope going through the ascender translates to a foot of travel up the rope
+ rope carries all of the load (not a problem for modern ropes)
+ lots of new technology and devices on the market, with more coming
+ The Akimbo! The Bulldog Bone! The Rope Runner!
+ multiple redirects don't add friction to the climbing system, as far as effort required to ascend/descend
+ redirects are easy and can help mitigate shock loading by spreading anchor point forces
+ rapidly evolved rope walking devices/techniques allow larger leg muscles to do most of the work ascending
+ less rope clutter, single line while ascending/descending and work positioning is easier to manage
+ using a base anchor is easy
- a base anchor seriously increases loading/shock forces at the TIP
- canopy anchors that you can install/retrieve from the ground takes a bit more thought and planning
- less rope in the system, between you and a canopy anchor, so shock loading is higher, requiring more awareness about fall mitigation
- new multicender devices are rather costly
This is certainly not a complete list... there are more pro and con items, but it gives you an idea about the many advantages/disadvantages of both techniques and systems.
Remember... a lot of this is highly subjective when it comes to real world climbing. Every climber will find that one system or the other is right for them for the way they prefer to climb. There is no law saying when and how you utilize the different methods, and experimenting with both usually yields the best system for whatever you are trying to accomplish at the time!
The fact is, some folks (myself included) will find that both systems are absolutely the best... depending on how you are going to climb that day, and the tasks you want to get done.
Hope that helps.