HMC allows you to jump quite a large distance by obeying the 'equations of motion' of the 'Hamiltonian'. This helps give more diverse samples, making them less correlated thereby increasing number the effective number of samples.
Right, but I mean that HMC has parameters to tune and multiples the computation needed per sample produced so it's not clear to me for which problems it's significantly beneficial and with how much tuning. I'd guess it's best where you otherwise don't have a good proposal distribution for Metropolis-Hastings leading to either many rejects or very slow mixing.
The NUTS sampler limits how much tuning you need. As far as I know it's currently one of the best algorithms for sampling the kind of posterior probability distributions you get with Bayesian methods.
It is indeed quite cool how it works.