Both are remasters of Age of Empires 2: Age of Kings. Definitive Edition is the second rerelease and, in addition to the higher quality graphics and all HD Edition content, includes updates and improvements. Some of the notable ones are:
- More civilizations from Eastern Europe & Central Asia (Lithuanians, Bulgarians, Cumans, & Tatars)
- Interface improvements such as a visible global production and research queue
- Improved AI
- Various quality of life features including automatic farm reseeding
- Server-based multiplayer
A full list can be found here: https://ageofempires.fandom.com/wiki/Age_of_Empires_II:_Definitive_Edition
Additionally, Definitive Edition still receives regular balance patches and content updates. HD edition is unlikely to receive more than compatibility updates.