Both are pretty versatile and make use of local markdown files. Logseq is more ouliner/bulleted note focused, while Obsidian is paragraph first (but with plugins for either you can really modify this quite a bit). Another difference is Obsidian organizes things into folders, while Logseq’s organization is flatter and more reliant on tags and hyperlinks to connect things (although you can nest pages, for instance having pages like this: pets, pets/cats, pets/dogs). Obsidian is more stable with a larger plugin ecosystem, but Logseq is being very rapidly developed and the dev team is super responsive.

Finally, Logseq is open source, while Obsidian is not. Their monetization models are pretty similar too, with the free version of both being really generous and limited features like Logseq/Obsidian-native Sync being available for a $5 monthly subscription. I regularly use both and encourage you to check them both out and explore what works for you.