First, DP are gen 4 games, and BDSP are gen 8 games. Here are the major differences between gen 4 and gen 5, gen 5 and gen 6, gen 6 and gen 7, and gen 7 and gen 8. Additionally, there is an incomplete list of modified moves that includes simple changes, like base power changes, but does not include stuff like defog removing hazards on the user's side in BDSP.
These are the changes in BDSP specifically. A lot of people are editing Bulbapedia right now, so this answer will probably become outdated soon. I recommend checking the Bulbapedia page again in 1 or 2 months.
Overall, the game's graphical style is different compared to Pokémon Sword and Shield. The camera returns to a mostly top-down viewing angle similar to the original Diamond and Pearl, while the player and NPCs appear with chibi proportions in the overworld but regular proportions in battle. The regular overworld uses random encounters only, while Pokémon Hideaways in the Grand Underground use symbol encounters.
Like in the originals, the games only include Pokémon from the first four generations; therefore, the games include a total of 493 Pokémon.
New features
The Underground has been replaced with the Grand Underground.
Pal Park has been replaced with Ramanas Park.
The Global Trade Station (GTS) has been replaced with the Global Wonder Station (GWS).
Amity Square now supports up to six walking Pokémon instead of just one. It retains the limitation that only "cute" Pokémon may walk, but it is unknown if this is defined the same way as the original games. (Only Pokémon that could walk in Amity Square in the original games have been seen so far, plus the three unevolved starters. The evolved forms of the starters are supported as well.)
The player may also pose for screenshots and save them to the Nintendo Switch's Album.
The Poffin-making minigame can now be performed inside Amity Square. The player's walking Pokémon surround the bowl during the minigame, visibly reacting to the player's cooking.
With the Nintendo Switch's lack of a bottom screen, the Pokétch may now be either absent from the screen, present in the top right corner of the screen, or enlarged to take up the majority of the screen (during which the regular overworld in the background has a blur filter applied). These three different methods of viewing are switched between using the R button. The touch screen or a pointer cursor can be used to control the Pokétch while it uses the enlarged screen.
HMs have been removed; instead, the Pokétch has a new "Hidden Moves" app allowing the player to call wild Pokémon to perform field moves corresponding to the original game's HMs. These wild Pokémon are the same regardless of location: Bidoof for Rock Smash and Cut, Staraptor for Fly and Defog, and Bibarel for Strength, Rock Climb, Surf, and Waterfall.
Hidden Moves now skip the splash screen animation when used multiple times in the same area.
It is implied that at least some touch features of the Pokétch are available even when not enlarged, as an off-center Dowsing Machine circle can be seen in one screenshot.
The Town Map now has new features allowing it to show the locations of Honey Trees (distinguishing between Honey Trees with no Honey, Honey Trees with Honey but no wild Pokémon, and Honey Trees with wild Pokémon) and Berries that can be picked.
Due to this, the Pokétch's Berry Searcher app has been removed, as it would be redundant.
The miniature sprites used for all Pokémon between Generation III and Sword and Shield now have a different appearance, no longer looking pixelated.
The icons that represent types have undergone numerous changes. For example, the plain circle icon for the Normal type now depicts two nubs resembling Bidoof's ears.
Pokémon now have different cries when they faint.
If the Nintendo Switch system being used has save data from previous mainline Pokémon games, the player may talk to an NPC in Floaroma Town to receive a Mythical Pokémon:
Jirachi, if Sword and Shield save data exists.
Mew, if Let's Go, Pikachu! and Let's Go, Eevee! save data exists.
Returning features
Seals return for the first time since HeartGold and SoulSilver, and are now called Stickers.
The player may now earn Stickers from completing Super Contest Shows or by defeating Gym Leaders.
There are now "over 90" different Stickers in the game, up from 77. "Ribbon Sticker C" and "Ribbon Sticker D" have been seen in screenshots as new Stickers.
Up to 20 Stickers may be placed on a Ball Capsule, up from 8.
The menu for positioning Stickers on a Ball Capsule now supports 3D movement for more detailed placement.
The boy in the easternmost house of Solaceon Town no longer awards letter Stickers for showing him the different forms of Unown, instead awarding extra Ball Capsules.
Pokémon Super Contests return for the first time since Platinum, and are renamed Super Contest Shows.
Ball Capsules and Stickers are now used during the Visual round in place of Accessories.
The points earned during the rounds are now named Hype Points.
An unspecified mechanic appears to involve adding up the Hype Points of all four competitors and comparing it to a target score.
Contests now support online connectivity.
The Union Room returns for the first time since Generation V.
There are now two rooms, called the "Local Room" and "Global Room", the latter of which newly supports online connections.
The Manaphy Egg is obtainable through an event for the first time since Generation IV. It is now an early-purchase bonus event.
TMs are single-use again, which was the case for all Pokémon games before Generation V.
Unlike in Diamond and Pearl, however, TMs have repeatable acquisition methods. Normal NPCs give three of the TM, while Gym Leaders give five of the reward TM with the badge. All TMs can also be bought in the Grand Underground by trading Spheres.
The National Pokédex returns in a core series game for the first time since Pokémon Omega Ruby and Alpha Sapphire (though it only features the first 493 Pokémon)
Honey Trees return for the first time since Platinum.
Footprints are displayed in the Pokédex for the first time since Pokémon Black 2 and White 2.
The Badge-polishing feature returns. The Badge case can be viewed from the Trainer Card.
The Key Items Vs. Seeker and Poké Radar both return.
The Moss Rock, Ice Rock, and special magnetic field have returned following their removal in Sword and Shield.
Several moves that were unusable in Sword and Shield have returned: Flash, Sketch, Hidden Power, Tail Glow, Psycho Boost, Heart Swap, Rock Climb, Chatter, Judgment, Dark Void, Seed Flare, and Toxic Thread. More information can be found here.
The Berry growth mechanic return after it was removed from Sword and Shield.
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