Authcache_panels vs panels_hash_cache

Which is recommended and under which circumstances?

In general if the site is content-centric, i.e. if the only thing which varies for users is a shopping cart, a list with recent comments or some personalized bookmarks, then try Authcache first. On the other hand it is not so well suited for user-centric sites, e.g. social networks where many pages directly depend on user-dependent parameters.

Also I tend to think that full page caching like Authcache and Varnish are not very well suited to speed up a slow site but merely to scale a decent one to a high volume. Just keep in mind that users will hit an uncached page from time to time.

I think I'll be asking this of many modules so that I can add to a 'howto'.

Great! I very much appreciate that.

authcache integrates with expire.
I assume thus authcache_panels would too.
Authcache_panels has the option of either using esi or ajax to render the panel pane.

Correct.

I assume this also applies to panelizer or anything that might utalize authcache_panels_page_manager.

Not quite. In order to make Panelizer panes loadable via Authcache Ajax/ESI it is necessary to write some glue code, i.e. a module similar to Authcache Panels by Page Manager. That module would have to collect the authcache settings of individual panes and report them to Authcache Personalization API (by implementing hook_authcache_p13n_fragment).

But I'm not sure how this is different from panels_hash_cache.

Note that the invalidation logic of personalized fragments (loaded via Ajax or ESI) is very simple: A users cache is cleared on every POST requested initiated by the respective user. Integration with the expiry module only affects the whole-page cache.