Thanks for the ping @oschulz. Here are some of my thoughts.
What’s AdvancedHMC.jl (AHMC) for
AHMC is motivated for two purposes.

Serve as the HMC backend of the Turing probabilistic programming language.
 It extracted the HMC codes from Turing.jl written during my MPhil in 2016 and becomes a standalone package on its own.

Serve as a research platform for developing novel variants of HMC algorithm
 We made some efforts so that we can quickly develop new HMC algorithms using AHMC. You might see some novel samplers available in AHMC shortly.
Note that because of these two purposes, the direct use of AHMC requires some basic understanding of HMC algorithms, and might not be suitable for users of all levels. But you can achieve more stuff with it because of its flexibility and GPU support. E.g. you can sample from an energybased model defined by a neural network using Flux.jl running on GPUs via CuArrays.jl.
See our poster on StanCon 2019 for more details of AHMC in which we also statistically compare our noUturn sampler (NUTS) implementation to Stan’s based on MCMCBenchmarks.jl.
The differences between AHMC and DynamicHMC.jl (DHMC)
The main difference is AHMC aims to support a wide range of HMC variants, including both static and dynmiac HMC algorithms, while DHMC, as the name indicates, implments a specific variant of dynamic HMC algorithm: NUTS with mutinomial sampling and generalised nouturn criterion; AHMC also supports this variant. Again our poster illustrates the variants that AHMC currently supports.
The Turing language and the differences in modelling pipeline
For endusers who focus on modelling, I highly recommend them to use Turing directly. The default HMC samplers are based on AHMC, but you can choose to use the NUTS implementation by DHMC as well. Plus there are many other inference methods available in Turing.jl, e.g. importance sampling, SMC, particle Gibbs, variational inference, etc. You will need some of those if you are working with discrete data
PS: A compositional interface that combines different MCMC samplers in a Gibbs algorithm is also available (see our AISTATS paper).
Turing has its modelling language (quite similar to Stan’s) which automatically uses Bijectors.jl (which is another library extracted from Turing.jl) to deal with constraint variables. The use of Turing.jl and Bijectors.jl (roughly) corresponds to the use of LogDensityProblems.jl and TransformVariables.jl.
For analysis, Turing provides MCMCChains.jl that implements a unified and easy way to check a lot of statistics for MCMC as well as plotting useful figures. This is also the default MCMC summary backend of CmdStan.jl.