What is the difference between PS1 and PROMPT_COMMAND?

PROMPT_COMMAND can contain ordinary Bash statements whereas the PS1 variable can also contain the special characters, such as '\h' for hostname, in the variable.

For example, here is my Bash prompt that uses both PROMPT_COMMAND and PS1. The Bash code in PROMPT_COMMAND works out what Git branch you might be in and displays that at the prompt, along with the exit status of the last run process, hostname and basename of the pwd.

The variable RET stores the return value of the last executed program. This is convenient to see if there was an error and the error code of the last program I ran in the terminal. Note the outer ' surrounding the entire PROMPT_COMMAND expression. It includes PS1 so that this variable is reevaluated each time the PROMPT_COMMAND variable is evaluated.

PROMPT_COMMAND='RET=$?;\
  BRANCH="";\
  ERRMSG="";\
  if [[ $RET != 0 ]]; then\
    ERRMSG=" $RET";\
  fi;\
  if git branch &>/dev/null; then\
    BRANCH=$(git branch 2>/dev/null | grep \* |  cut -d " " -f 2);\
  fi;
PS1="$GREEN\u@\h $BLUE\W $CYAN$BRANCH$RED$ERRMSG \$ $LIGHT_GRAY";'

Example output looks like this in a non-Git directory:

sashan@dhcp-au-122 Documents  $ false
sashan@dhcp-au-122 Documents  1 $

And in a Git directory you see the branch name:

sashan@dhcp-au-122 rework mybranch $

Update

After reading the comments and Bob's answer, I think that writing it as he describes is better. It's more maintainable than what I originally wrote above, where the PS1 variable is set inside the PROMPT_COMMAND, which itself is a super complicated string that is evaluated at runtime by Bash.

It works, but it's more complicated than it needs to be. To be fair, I wrote that PROMPT_COMMAND for myself about 10 years ago and it worked and didn't think too much about it.

For those curious as to how I've amended my things, I've basically put the code for the PROMPT_COMMAND in a separate file (as Bob described) and then echo the string that I intend to be PS1:

GREEN="\[\033[0;32m\]"
CYAN="\[\033[0;36m\]"
RED="\[\033[0;31m\]"
PURPLE="\[\033[0;35m\]"
BROWN="\[\033[0;33m\]"
LIGHT_GRAY="\[\033[0;37m\]"
LIGHT_BLUE="\[\033[1;34m\]"
LIGHT_GREEN="\[\033[1;32m\]"
LIGHT_CYAN="\[\033[1;36m\]"
LIGHT_RED="\[\033[1;31m\]"
LIGHT_PURPLE="\[\033[1;35m\]"
YELLOW="\[\033[1;33m\]"
WHITE="\[\033[1;37m\]"
RESTORE="\[\033[0m\]" #0m restores to the terminal's default colour

if [ -z $SCHROOT_CHROOT_NAME ]; then
    SCHROOT_CHROOT_NAME=" "
fi
BRANCH=""
ERRMSG=""
RET=$1
if [[ $RET != 0 ]]; then
    ERRMSG=" $RET"
fi
if which git &>/dev/null; then
    BRANCH=$(git branch 2>/dev/null | grep \* |  cut -d " " -f 2)
else
    BRANCH="(git not installed)"
fi
echo "${GREEN}\u@\h${SCHROOT_CHROOT_NAME}${BLUE}\w \
${CYAN}${BRANCH}${RED}${ERRMSG} \$ $RESTORE"

And in my .bashrc file:

function prompt_command {
    RET=$?
    export PS1=$(~/.bash_prompt_command $RET)
}
PROMPT_DIRTRIM=3
export PROMPT_COMMAND=prompt_command

From the GNU Bash documentation page (Bash Reference Manual):

PROMPT_COMMAND
    If set, the value is interpreted as a command to execute before
    the printing of each primary prompt ($PS1).

I never used it, but I could have used this back when I only had sh.