Polymorph and Wild Shape are similar but different. Both allow a character to transform entirely into a different creature. However, there are distinct differences.
First of all, Polymorph allows a creature to transform into much higher level (CR) creatures. It allows one transform into a creature (beasts only) = to your character level. Wild Shape (Moon circle only – PHB 69) allows you to transform into your character level divided by 3. Big difference in the type of creature! For example, at character level 8, a druid can transform into something like a Rhino (CR 2), while polymorph can transform into… a T-Rex (CR 8)!!!
On the other hand, Wild Shape has more finesse with its abilities. Eventually a character can cast spells while Wild Shaped (PHB 67). The other major difference? Wild shape keeps a character’s mental abilities (INT, WIS, CHA) [remember spell saves]. Polymorph makes the character as dumb as the beast they are transformed into.
Pro-tip: Casting find familiar first, then Wild Shape, allows a character to cast touch spells via their familiar very early in the game, instead of waiting until level 18
Pro-Pro Tip: True Polymorph allows a character to transform into any creature, not just a beast…like a dragon…