You can make the whole script a function like this:

main_function
{
do_things_here
}

then at the end of the script have this:

if [ -z $TERM ]
then
# if not run via terminal, log everything into a log file
main_function 2>&1 >> /var/log/my_uber_script.log
else
# run via terminal, only output to screen
main_function
fi

Alternatively, you may log everything into logfile each run and still output it to stdout by simply doing:

# log everything, but also output to stdout
main_function 2>&1 | tee -a /var/log/my_uber_script.log

You can make the whole script a function like this:

main_function() {
  do_things_here
}

then at the end of the script have this:

if [ -z $TERM ]; then
  # if not run via terminal, log everything into a log file
  main_function 2>&1 >> /var/log/my_uber_script.log
else
  # run via terminal, only output to screen
  main_function
fi

Alternatively, you may log everything into logfile each run and still output it to stdout by simply doing:

# log everything, but also output to stdout
main_function 2>&1 | tee -a /var/log/my_uber_script.log