I think the reason I liked it so much (and this also applies to URF/ARURF) is that it is a mode where you could pick any champion, absolutely any champion you desire, and go anywhere. You don't have to worry (as much) about lane, role, what is good, etc. SR doesn't have that really. Now that TT has been lost, which was the closest mode there was to offering this, it hurts (the bots just moved to co-op vs. A.I., the OTHER mode that offers this). As a mode, neither NB or URF is really "better" than SR... but at the same time, you can jump in queue, pick whoever, and off you go... and that appeals to me these days, all in the time an SR lobby is abandoned because someone dodged halfway through pick.
ARAM is still great for what it is, but it differs in that it's a one lane teamfight from the start, and you can't pick your champion.
I remember thinking I'd like the very original URF, only to dislike it after two or three rounds. When ARURF came back around last Christmas, I fell in love with it, especially as it came right after NB. I went from not playing League in like 3 years to playing nearly nonstop those few months. I'd literally be playing a lot if either mode came back.