Splitting this into two parts:
Grappling
PHB14 (p. 195) has a few paragraphs detailing how grapples used to work, as well as several restrictions on how they couldn’t work. It stipulates that the attacker has to have one free hand, that the target can only be a single size larger than the attacker, and the target has to be within the attacker’s reach. The attempted grappler uses their Attack action (consuming one of their multi-attacks in the process) and the attacker and target make contested rolls to determine whether the attempt is successful.
The '24 PHB changes a few things up. First of all, it - confusingly - splits the rules across two different Glossary entries 10 pages apart (p. 367, 377).
Page 367 (Glossary entry: “Grappling”) specifies that the attacker attempting a grapple makes an Unarmed Strike with whatever body part it has appropriate to the task, and that escaping the grapple requires using an Action to make an Ability Check (Strength or Dexterity). Later in the Glossary, the “Unarmed Strike” (p. 377) entry explains that resisting the grapple attempt is no longer a contested roll, but that the target makes a saving throw, their choice of Strength or Dexterity, at DC 8 + the attacker’s Strength modifier and Proficiency Bonus. Size and available appendage restrictions remain.
There are a some interesting wrinkles here. First off, PHB24’s “Unarmed Strike” entry generally states that a hit inflicts damage equal to 1 + the attacker’s Strength modifier - and grapples are not explicitly excluded from this! If this is indeed the case (Edit: it’s not, I just missed it), then it seems foolish not to exploit this nearly all of the time and just make every unarmed attack a grapple attempt.
The text also specifies that avoiding the initial grapple is a saving throw, but attempts to escape a grapple are an ability check - this is a bit of an awkward phase-change; it is logical, but not at all intuitive so it will absolutely lead to the wrong ruling in play.
That covers the mechanics, but the condition itself has some changes, too. Appendix A: Conditions (PHB14, p. 290) details the previous condition, and the PHB24 Glossary (p. 367) has the updated terms. The change here can be tricky to parse because '14 lists some mechanics covered elsewhere, but it amounts to this: a grappled creature has Disadvantage on attacks against anything other than its grappler.
Lastly, PHB14 stipulated that the grappler’s “speed is halved”, while PHB24 says “every foot of movement costs…1 extra foot” based on the size differential. Both books agree that this penalty doesn’t apply to when the grappler is two sizes larger than the target, but PHB24 goes a (half) step further and negates any movement penalty when the target is Tiny.
Shove
Thankfully this section is less involved. PHB14 (pp. 195-196) employed the contested roll approach and PHB24 (p. 367) makes this a saving throw instead. As with grapple, PHB24 appears to apply damage to the shoving attack. Technically speaking, PHB24 also imposes the Prone condition, but this is not functionally different that PHB14’s “knock the target prone” language.