What I love about Picasa is these gems that I found nowhere else:
indexing my own folder structure
indexing of defined folders only + watching for updates
printing possibilities: layout, border etc. The whole printing interface is very clever
annotations to images, starring etc
creating albums
batch export function (for albums or selected images from the stack) + export option directly to mail as attachment
the interface concept in general
quick image-editing (contrast, lightness, retouch, crop, adjust angle)
So basically, almost everything :D
I would so much love to see someone resurrect the concept of what picasa provided. I still hope for Affinity :)