Oh man, this is SO MUCH better than old Grappler!

Grapple builds are so back 🙏

On monks this feat is insanely strong. Run in, grapple while damaging, have advantage on multiple unarmed strikes in a row and absolutely wreck your enemies. Drag enemies away from your allies for miles with step of the wind and fast wrestler, preferably into dangerous terrain/spell AOEs or off of high elevations. You're also pummeling them with advantage while they have disadvantage to hit anyone else.

In other words, we're so back grapple bros!

This feat is great for monks but it takes the Open Hand Monks to another level especially by level 11. At level 11, an Open Hand Monk with this feat can potentially in one turn:

• Grapple a creature of its size or smaller and do unarmed strike damage with that attack still. 
• (Assuming the grapple is successful) attack them again with advantage with their second attack of their attack action. If you've acquired a weapon mastery (through multiclassing or the feat) a quarterstaff could be used to topple them or (if you really want to be petty) you can leave the toppling to your Open Hand Techniques and instead use a Spear to apply Slow.
• If the first attacks grapple wasn't successful then you make a second unarmed strike but try again for the grapple. After this second attempt I'm going to assume they are grappled.
• Flurry of blows them to make (thanks to Heightened Focus) three unarmed strikes against them with advantage (assuming they are grappled) as a bonus action. Gives plenty of opportunity to give that enemy Addle and a further opportunity to make them prone while dealing damage (thanks to Open Hand Technique).
• As part of that same bonus action use Step of the Wind (and Fleet Step) to double your speed
• Drag that enemy 50 feet away from combat into another room that you've cleared, leave them there (which helps split enemies up and makes each one more vulnerable to your party swarming the rest of the enemies) and run back to the combat (and close the door on them if you can).

Thanks to addle they can't even make an opportunity attack against you when you leave them and, thanks to the Open Hand's topple, they are also likely prone (and may have Slow) and won't get far next turn. This whole thing cost only one ki point.

Rinse and repeat as you need to split up enemies. You can also use your Open Hand Techniques push and extra movement to keep them in that room too.

You can also drag them up a vertical surface and drop them but those opportunities don't always exist. You may also use Stunning Strike on them at some point too.

Of course the above is subject to chance like the rest of the game, but this single feat gives them so much more flexibility for dealing with enemies.

Punch and Grab. is quite strangely written.

Unarmed Strike is a aggregate rule that has 3 options and only one of them needs to hit (other 2 are Save based). For me this is confusing if I should consider Grapple and Shove as hits. They are Attacks, but they do not make an Attack Roll (spells are not Attacks actions, they are Magic actions). So can I pick Grapple as my main Unarmed Strike option and then just deal Damage as a free bonus here? So to bypass a high AC but low Save? Probably not.

It still is a bit confusing in the part "do I have to hit AND my target makes a Save against Grapple" OR "if I hit with Damage my target is automatically Grappled"?

Also I guess that Shove is not part of this Feat at all.

So I understand this Feat as: "Punch and Grab.. When you hit a creature with an Unarmed Strike(Damage) as part of the Attack action on your turn, you can use both the Damage and the Grapple option. The target can make a separate Save against being Grappled. You can use this benefit only once per turn."

This makes sense but also weakens the Feat a bit. If I have 70% to hit with an Attack Roll but my opponent has a 60% change of making a Save (most likely as it can choose Str or Dex) then I have only around 30% chance of both option succeeding - Damage and Grapple. Also I have to hit with Damage first, for the Grapple Save to occur, so they are dependent.

@JakubW: I think that if you had to roll to hit AC and do damage before using the Grab part, then the rule would say "When you damage a creature with an Unarmed Strike." A way to consider the grappling rules is that a grapple attempt auto-hits, since the attacker doesn't roll to hit. It's up to the target to save. I think it's a little sloppily written, and am reading it as "When you take an Unarmed Strike as part of the Attack action on your turn..." because that's the language that would have provided the greatest clarity.

I actually came to the comments wondering a different question: Can you choose to do damage and to grapple in any order? Instead of attacking and then doing a grapple check, can I force a save against my grapple DC first, and then roll to attack with Advantage if they fail?

@SpcAgentOrange

To be honest, I think that your question is the same as my question. And it is:
"Is Unarmed Strike(Grapple) and  Unarmed Strike(Shove) an attack that hit or miss?"
If the answer is "yes" then I think that you can do what you propose. There is no problem with that as you
first "When you hit a creature with an Unarmed Strike(Grapple)"
then: "you can use both the Damage and the Grapple option."

But the main problem is the question
1. "Does an Attack that does not roll an Attack Roll but forces a Save Roll is considered a hit/miss situation"
OR
2. "regardless of it being an Attack it is treated as a Spell and does not hit/miss, but the opponent Saves or not".

If 2 is true then you have to play this as I've outlined as you cannot HIT with "When you hit a creature with an Unarmed Strike(Grapple)". You can only do that with "Unarmed Strike(Damage)".

But if 1 is true then you should even be able to HIT with Unarmed Strike(Shove) and then use other 2 options. Or as you've outlined first do Unarmed Strike(Grapple) and then do Unarmed Strike(Damage) with Advantage.