Okay, this is actually a question I see pop up a lot, and having read all the manga volumes and re-watched the anime a few times, I think the core difference boils down to how the story is paced and what gets prioritized. The anime, being a single 12-episode season, has to condense a ton. It covers roughly the first four volumes, but it does so by sprinting through key plot beats, especially the early 'capture the candidate' arcs. You get the major events, but a lot of the smaller character moments and world-building tidbits from the manga get left on the cutting room floor.
Where it diverges most noticeably, though, is the ending. The anime crafts an original finale to provide some closure for the season. It uses elements from later in the manga, specifically the 'Ciel' storyline, but rearranges them into a self-contained boss fight. In the manga, that whole section is way more complex, with different motivations and a much slower reveal of certain truths about the Demon Lords and Arata's past. The anime's ending feels a bit more... conventional shonen finale, while the manga keeps layering on mysteries.
Also, the fanservice balance is different. The manga has it, obviously, but it's woven into the comedy and plot more evenly. The anime sometimes feels like it amplifies those moments for, well, anime reasons. You lose some of the more clever visual gags from the manga panels in favor of more overt scenes. The characterization of the girls, especially someone like Lilith, feels slightly flatter in the anime; the manga gives them more room to breathe and show their personalities outside of just 'harem member.' I'd say the anime is a fun, sparkly trailer, but the manga is the full, convoluted, and surprisingly engaging novel it's based on.