What's the difference between "Export as: Jpeg or do "Save for Web (Legacy)""

Thanks for all your replies! That was so clear and helpful. On quality: would you choose nearest neighbor, bicubic, etc? Which one of those would be my most used option? Thank you again.

There are still shortcomings in Export As that basically makes it useless. The most serious one is that it cannot be used in actions. It just doesn't record. That's a dealbreaker.

I also don't understand why "embed color profile" is still not checked by default. At the very least it should stick and stay checked once you do it - but you have to remember to check it every single time. It's ridiculous. Yes, I realize there are still situations where an embedded profile is not needed or irrelevant, but in that case it does no harm. It's just ignored anyway. And no, don't tell me about file sizes. It's 3 kilobytes.

The main argument for keeping it checked is very simple: it matches Photoshop. The initial mismatch between Export/Save For Web and Photoshop causes confusion and damage no end.

Save for Web has been around for so many years that the code is getting old and creaky. Large documents take forever to load into it. Save for Web also does not handle current web standards such as 2x/3x scale factor export.

Export As is new code with more up-to-date capabilities for web/mobile graphics production. But it isn’t fully completed; it has less control over metadata and can’t save an optimized animated GIF. Until Export As can do everything Save for Web does, Save for Web will be kept around. But Adobe added (Legacy) to the name to let everyone know that it is on the way out eventually.

Note that there are two Export As commands that do different things. File > Export > Export As exports entire artboards; that’s why it’s on the File menu.  Layer > Export As exports individual layers to separate files, so it’s capable of generating layer-based assets for web/mobile developers to integrate into projects.

Export as will eventually replace save for web, but they're still refining it. It does or will have better saving quality.

In Save for Web (legacy) dialogue you have more/additional options and finer control plus it can strip unecessary data for web. Otherwise you should not see difference in quality of image if equal options and quality is chosen.

Update: I missed point, in my answer Save for Web and Save As are compared.