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What, precisely, is the difference in what is being performed by mkdir -p and install -d, in terms of what changes the two commands are doing to the system?

For starters, mkdir -p is POSIX, install is not. Then, we have this from the GNU install documentation:

If the --directory (-d) option is given, install creates each
directory and any missing parent directories. Parent directories are created with mode ‘u=rwx,go=rx’ (755), regardless of the -m option
or the current umask. See Directory Setuid and Setgid, for how
the set-user-ID and set-group-ID bits of parent directories are
inherited.

And:

‘-d’

‘--directory’

Create any missing parent directories, giving them the default
attributes. Then create each given directory, setting their owner,
group and mode as given on the command line or to the defaults.

So:

1. For GNU install, the permissions of the parent directories could be different.


2. 
   install lets you set the ownership of the leaf directory.

The main difference between mkdir -p and install -d is that if the directory already exists, only install -d will try to set the ownership and file mode bits.

• 
  install -d will call make_dir_parents with preserve_existing set to false
  


• 
  mkdir -p will call make_dir_parents with preserve_existing set to true.

If preserve_existing is true and the directory already exists, the function will not attempt to set the ownership and file mode bits.

mkdir -p and install -d in coreutils call the exact same make_dir_parents function.

Coreutil sources:

• mkdir.c


• install.c