Why does an SSH remote command get fewer environment variables then when run manually?

There are different types of shells. The SSH command execution shell is a non-interactive shell, whereas your normal shell is either a login shell or an interactive shell. Description follows, from man bash:

       A  login  shell  is  one whose first character of argument
       zero is a -, or one started with the --login option.

       An interactive shell is  one  started  without  non-option
       arguments  and  without the -c option whose standard input
       and error are both connected to terminals  (as  determined
       by  isatty(3)), or one started with the -i option.  PS1 is
       set and $- includes i if bash is interactive,  allowing  a
       shell script or a startup file to test this state.

       The  following  paragraphs  describe how bash executes its
       startup files.  If any of the files exist  but  cannot  be
       read,  bash reports an error.  Tildes are expanded in file
       names as described below  under  Tilde  Expansion  in  the
       EXPANSION section.

       When  bash is invoked as an interactive login shell, or as
       a non-interactive shell with the --login option, it  first
       reads and executes commands from the file /etc/profile, if
       that file exists.  After reading that file, it  looks  for
       ~/.bash_profile,  ~/.bash_login,  and  ~/.profile, in that
       order, and reads and executes commands from the first  one
       that  exists  and is readable.  The --noprofile option may
       be used when the shell is started to inhibit  this  behav­
       ior.

       When a login shell exits, bash reads and executes commands
       from the file ~/.bash_logout, if it exists.

       When an interactive shell that is not  a  login  shell  is
       started,  bash reads and executes commands from ~/.bashrc,
       if that file exists.  This may be inhibited by  using  the
       --norc  option.   The --rcfile file option will force bash
       to  read  and  execute  commands  from  file  instead   of
       ~/.bashrc.

       When  bash  is  started  non-interactively, to run a shell
       script, for example, it looks for the variable BASH_ENV in
       the  environment,  expands  its value if it appears there,
       and uses the expanded value as the name of a file to  read
       and  execute.   Bash  behaves  as if the following command
       were executed:
              if [ -n "$BASH_ENV" ]; then . "$BASH_ENV"; fi
       but the value of the PATH variable is not used  to  search
       for the file name.

How about sourcing the profile before running the command?

ssh user@host "source /etc/profile; /path/script.sh"

You might find it best to change that to ~/.bash_profile, ~/.bashrc, or whatever.

(As here (linuxquestions.org))

Shell environment does not load when running remote ssh command. You can edit ssh environment file:

vi ~/.ssh/environment

Its format is:

VAR1=VALUE1
VAR2=VALUE2

Also, check sshd configuration for PermitUserEnvironment=yes option.