Persona 5 Royal is similar to Persona 3 FES and Persona 4 The Golden: it's an enhanced version of the base game but with a sequel included (except for Persona 4 The Golden, which doesn't add a sequel).
The most important changes at a glance:
A sequel in the form of a third semester. The third semester can only be accessed if you managed to meet the requirements before November 17.
A new playable character and confidant: Yoshizawa Kasumi of the Faith Arcana.
A new confidant: Takuto Maruki of the Councilor Arcana.
A new supporting character: Jose.
The story has been slightly tweaked to accommodate for the additional characters and the new third semester.
Additional changes include:
4K 30 FPS support on the PS4 Pro.
The "Thieves Den" as the protagonist's palace. It is a recreational area unrelated to the story.
New songs.
Slightly different Playstation Trophies.
New costumes.
New dialogue portraits.
Lip sync during voiced dialogue scenes (Japanese voiceover only).
Some unvoiced scenes are now fully voiced.
Additional scenes (from Persona 5 The Animation).
Some characters have new voice actors.
A scene (where one character is being harassed by two men) has had its dialogue slightly modified for the English localization.
The international release now has Japanese audio included (was a DLC before).
The international release now has French, Italian, German, and Spanish subtitles.
A new font to support the additional languages. Some names have been updated as a result, e.g. "Arsène" and "Café Leblanc".
School Life
A new location in Tokyo that can be visited: Kichijoji.
A couple new hangouts: An aquarium in Shinagawa, and the Nakano shopping district.
New activities, such as darts, billiard, or drinking.
The protagonist can increase his max SP by visiting a temple.
A clothing shop has been added, where the protagonist can buy new outfits.
The Justice Arcana is no longer automatic.
Existing confidant abilities have been modified, and can grant new perks.
The protagonist can hang out with Caroline and Justine to gain Skill Cards.
The protagonist can now give gifts to his male confidants as well.
The protagonist's free time has been slightly rearranged.
It takes longer to improve the Guts and Kindness Social Stats.
Most of the daily activities have been modified or expanded.
There's a new "Daily Assist Command" to help the player during day time.
The Valentine's Day event has been updated with additional content.
There is now a White Day event.
Phantom Life
New Personas.
Some Personas and Shadows have their attributes, move sets, or Arcanas altered.
Party members have their skill pool slightly tweaked.
New skills, and the effects of some existing skills have changed.
New enemies.
Accessories can grant active skills in addition to passive effects.
A new palace for the third semester.
A new area in Mementos.
New ultimate Personas for the third semester.
Akechi returns as a playable character
New Mementos Requests for the third semester.
The Velvet Room can sometimes enter a Fusion Alarm state, which allows for the fusion of more powerful Personas, but increases the risks of fusion accidents.
A new Challenge Battle mode, where the player can battle various enemies. The protagonists from Persona 3 and Persona 4 are available as DLC boss fights.
Merciless difficulty is now included (was a DLC before).
Difficulties have been tweaked.
Unison attacks, called: "Showtimes".
New types of Shadows: Savage Shadows and Disaster Shadows.
New collectibles, called: "Will Seeds".
There is a rare chance to stumble upon "Deviations", special areas with different layouts within Mementos.
A Grappling Hook.
Futaba has her own Finishing Touch, a special animation that plays after an All-Out Attack defeats all remaining enemies.
The behavior of all major bosses has been altered or expanded.
Lavenza can be fought in Mementos during a New Game Plus.
Baton Pass is unlocked for all party members from the beginning. There is now a Baton Pass Rank instead. When maxed out, the Baton Pass is now more powerful than before.
The party members' Shadow Negotiation ability (during Hold Ups) now always works, but only under specific circumstances, which differ for each party member.
Technical attacks (attacking enemies afflicted by a certain status condition with certain elements) can be ranked up by playing billiard and has a chance to down enemies.
New status-element combos for Technical attacks, as well as the ability to learn combos from books.
Guns reload after every battle. You are no longer limited to a certain amount of ammo every time you visit a palace.
When frozen, enemies temporarily lose resistance to Physical and Gun skills.
Some status ailments now end upon getting hit. Additionally, Confusion ends 1 turn earlier than other status ailments.
The Reaper is now immune to the Despair ailment, and is therefore no longer vulnerable during Flu Season.
The Security Meter in palaces now starts at 40% instead of 0%.
Treasure Demons can now appear among random Shadow guards.