Quick disclaimer: I'm the author of Tini (thanks for the hat tip, by the way!).
Note that for interactive usage, Tini actually hands over the tty (if there is one) to the child, so in that case signals that come "from the TTY" (though in a Docker environment this is an over-simplication) actually bypass Tini and are sent to the child directly. This should include SIGSTP, though I'm not sure I tested this specifically.
That being said, both tools are probably indeed very similar — after all there is little flexibility in that kind of tool! Process group behavior is probably indeed where they differ the most. : )
Note that for interactive usage, Tini actually hands over the tty (if there is one) to the child, so in that case signals that come "from the TTY" (though in a Docker environment this is an over-simplication) actually bypass Tini and are sent to the child directly. This should include SIGSTP, though I'm not sure I tested this specifically.
That being said, both tools are probably indeed very similar — after all there is little flexibility in that kind of tool! Process group behavior is probably indeed where they differ the most. : )