Put another way, when is a grapple(MM) not a grapple(PHB)?
My wild shaped Circle of the Moon druid loves to be a giant crocodile. Big and strong, he can grapple much bigger creatures than our barbarian can. He is huge, so he can grapple Huge and larger creatures.
Bite. Melee Weapon Attack: +8 to hit, reach 5 ft., one target. Hit: 21 (3d10 + 5) piercing damage, and the target is grappled (escape DC 16). Until this grapple ends, the target is restrained, and the crocodile can't bite another target. (MM, Giant Crocodile)
So where does this DC of 16 come from to keep the grapple?
8 + (strength bonus~5+ + (CR 5 proficiency bonus~3) = 16.
(It is of interest that the giant crocodile is not listed as having the Athletics ability the way that a giant does.)
Druid is not polymorphed; he is wild shaped.
The Wild Shape feature says:
While you are transformed, the following rules apply:
Your game statistics are replaced by the statistics of the beast, but you retain your alignment, personality, and Intelligence, Wisdom, and Charisma scores. You also retain all of your skill and saving throw proficiencies, in addition to gaining those of the creature. If the creature has the same proficiency as you and the bonus in its stat block is higher than yours, use the creature’s bonus instead of yours. If the creature has any legendary or lair actions, you can’t use them. (PHB, Wild Shape).
My druid has Athletics proficiency(from outlander), and took a level in rogue to get Athletics expertise. His proficiency bonus is 4, which doubles to 8 when grappling or shoving in druid form.
Should I double the proficiency for the giant crocodile and compute the DC as 19, rather than 16, since I retain my abilities and proficiencies? (Double the +3, not the +4, in this case).
8+5+3+3 = 19
Or, is this a case of "that grapple isn't the same as the other grapple" even though the same term is being used? The reason for my uncertainty is that a PHB grapple does not, by default, restrain a creature (that takes the grappler feat), while certain monsters restrain a creature when they grapple them. (Ropers, for example).