If an artists posts a high resolution image to Twitter that's highly compressed and also posts the same image as a PNG to Tumblr but at a smaller resolution, which one should be posted/kept on e621? Or do they both stay for the sake of archiving?

Only one stays, resolution doesn't matter if the image is compressed af.

Open the images in two different tabs and start zooming in to compare. Post the one which is clearer and has less compression artifacts.

If you want to be more thorough, use an image editing software to compare the differences of the images in two layers.

TheGreatWolfgang pretty much summed it up. Altough browsers apply filtering when scaling content for nicer user experience, so I would suggest using local image editor/viewer for this which doesn't filter the content when zoomed.

PNG is always preferred because it's lossless, but if the resolution increase is massive, at that point slight compression doesn't matter that much. (Of course best resolution PNG would be most preferred, but sometimes these cases happen) Let's say Tumblr is restricted to 500x500 and twitter 2000x2000, that's already 16 times more pixels so even if the compression destroys some, there's still much more visible information. Then there's also way that the image was downscaled as sometimes the downscale can be really smooth, which at some cases can be bit more visually pleasing, but most of the time it's just worse. That's one reason why twitter really often is much better than FA even with just slight resolution increase.

That all said, there are cases where it's really hard to determine which one is actually better, in which cases PNG is preferred one and always put the other possible sources in source field as well so that users are free to compare it themselves. I like having transparency on these kind of things, which is why I sometimes link comparisons for example if I'm replacing video content on the site.