In a handstand and shoulder pops your arms are next to your head, so you're strong in that position. If you're not able to push through your shoulders in your handspring it's because A) you have too much force coming onto your shoulders for your current strength level to hold or B) you're not in a handstand shape.
Something that happens with a lot of new handspringers is that they let their feet come over the top of the handspring too quickly so when their hands hit the ground their feet have already come past vertical. When this happens it is impossible to get in a handstand shape and you'll have to create a shoulder angle. Angles other than fully extended are (usually, and by usually I mean almost always) weaker.
What you want is to be in a tight arch/extended position when your hands hit the floor in your back handspring.
To work on this do a handspring with your hands 2-3 feet from a wall (depending on your height). Face away from the wall and squeeze your back and bottom when your feet are on the wall. Feel the extended shoulder position that you want in your handspring. Keep your shoulders next to your ears and look down the length of your arms to your hands.
Then, using your abdominal strength pull your feet off the wall (AT THE SAME TIME) to get to a free handstand. If at home or without a mat just feel free to then step down, but if you have a 4in or 8in mat, keep your shoulders pushing extended and everything squeezed, and fall to your stomach in a tight hollow.
Doing back handspring up to an 8in to handstand and then falling to your stomach also help but work better with experienced handspringers unless a coach can assist.
