It was sheer and glorious beauty my friend. Of course, I'm assuming you know this and don't, but it was the SEC version of Alabama-Penn State. A lot of times you have Alabama fans whose "second favorite team" was Tennessee. It was like the civil war (not literally) in that two brothers would have a three-hour fist fight, the loser would raise the winner's hand, and they'd go have a beer afterwards. When Auburn beat us, there was resentment. When Tennessee beat us, there was respect.
It was all a rivalry SHOULD be. It's what the Iron Bowl SHOULD be. The only thing that's been kind of similar lately is when the two Mississippi schools were ranked in the top five in 2014, and both were publicly rooting for each other (whatever was said behind the scenes).
Coach Bryant respected Coach Neyland. His beef with Tennessee (small by comparison) might be because former Tide player Bill Battle was the Vols coach from 1970-76, and even though he never had a losing season and there were issues beyond his control, they fired him (granted, he was regressing record-wise). After all, Bryant would have had a thing with Tennessee going back through his time as Kentucky coach and on back to when he played as "the other end" to Don Hutson.
It's gone, and it's never coming back. If Phil Fulmer would have beaten us on the up and up, it would be a gallant rivalry of respect. Hell, they beat us four years in a row (1982-85) and nary a word of protest from us. They were the better team at least three of those years. But Fulmer knew his job security depended on beating Alabama at least every so often (he'd watched Battle and Majors both get canned, hell, he stuck the knife in Majors's back and told him it was a massage).
Incidentally, much of the bad blood between Florida and Tennessee (aside from their division rivalry) stems from the Vols voting in 1985 to strip Florida of their first-ever SEC title. The Gators DID win it on the field, but they opted to forego the Sugar Bowl, knowing severe sanctions were coming. In May 1985, the SEC had a vote and it was a 6-4 vote to strip Florida of the title and not recognize a champion in 1984. Florida, MSU, Alabama, and Auburn all voted to permit Florida to keep their crown. Tennessee voted against them, and given some of the crossover between the two schools (sort of like us with Clemson or ATM), this was considered a betrayal. After all, a Vols vote would have given Florida their first SEC title with a 5-5 decision (or at least forced it on Boyd McWhorter).