The action cost is different, but the effect is the same

Shield Master lets you shove without sacrificing an attack

Normally, when a character takes the Attack action, they can replace any of the attacks they get with a Shove action. Shoving in this way involves the sacrifice of one attack.

With Shield Master, the character need not sacrifice any of their attacks when they take an Attack action. Instead, they can shove using a bonus action.

They are very different things. A character who gets only one attack as part of an Attack action must give up all their attacks to shove someone. That is a significant sacrifice. Getting the same action as a bonus action means that one need not make that sacrifice. Which is a distinct advantage if that is something that the character is doing on a regular basis.

Effects of the shove are the same

Regardless of the actions used to take it, the character is still taking the Shove action. As such, the effects of the Shove are exactly the same.

Also, note that having Shield Master does not prevent you from taking the normal shove action (replacing one of your attacks) it just gives you and additional option. You could even do both in the same turn if you wanted (including shoving even before you attack).

Bonus actions are valuable

If you are playing a character that makes heavy use of their bonus actions for things other than Shield Master, then this part of Shield Master is probably not going to be the most useful to you. You only get one bonus action per turn and it is good to spend it wisely. However, not all characters do have a bonus action ability that they use commonly. In this case, if one wants to shove, this allows that character to put that bonus action to good use. In the end it all comes down to the specific character build you have/want to have.

The restrictions for using the Feat are clear:

• You must have taken the Attack action.
• You must have attacked only with glaive/halberd/quarterstaff.

The first bullet is easy-go: yes, you took the Attack action. The second bullet is probably a no, since you have attacked (as Shove is "a special melee attack") with something else that does not involve your quarterstaff (per Crawford below).

In a similar question, which asked

Does Shove qualify you to use the bonus attack in Two Weapon Fighting or Martial Arts?

Crawford answered

The shove and grapple options don't involve an attack with a weapon or an unarmed strike, so no. ¹

This means that both your interpretations in the "only" part lead to the same thing.

it's unclear if it mean you can't attack with other weapons, or that you can't take any actions other than polearm attacks

You are both "attacking with other weapons (where the other weapon is, probably, your body)" and "taking other (than polearm attacks) action".

¹ My interpretation from that is that No, you don't use your quarterstaff for the shove action. The reasoning for that is that if it does not involve an attack with a weapon, it either does not involve an attack (which is false, since it does) or does not involve a weapon (which is the only possible solution). It doesn't require a free hand because you can use your body, you don't need to push the person with your hands.