In answering the question to your thread title; everything. Simply put, there was nothing awful or bad about the old Tag Team division. It had everything from multiple top Tag teams that were to be viewed as the best and the most elite.. to the up-and-coming teams to look out for.. to the bottom of the ladder, jobber tag teams.. yet even those teams were viewed as nice to see and watch. Here is just a few of the teams I recall, from each category.
Elite:
1. Edge & Christian: Without question one of the absolute greatest Tag Teams to pay attention to during this stage in Wrestling history. They had the speed and electric-ness of a Hart Foundation/Rockers.. yet they had the strategy and technical ability of a team like Arn & Ole, DiBiase & I.R.S, and Steiners.
E&C as they were known were arguably the greatest reason the Tag Team division throughout 1998-2001 was the most loved and remembered time during Tag Team Wrestling. They held the titles some odd 7-8 times.. and trailed just behind a team like the Dudleyz (for most overall) and Harlem Heat.
2. The Hardys: While Edge & Christian were exciting to watch and well remembered, you can't discount that it was the Hardys who helped make Edge and Christian famous.. why? Well, because just like the saying goes.. you can't have peanut butter without jelly. The Hardys were equals in many ways to E&C, and helped each other become greatly remember able.
You can argue until you're blue in the face over who the masters of the T.L.C and Ladder matches were. Edge, Christian.. or Matt & Jeff Hardy. All four men used the Ladders and Chairs to their advantage, and made themselves great Tag teams, as well as single's stars because of it. The Hardys were a great team to watch during this period, not just because of their chemistry, but because they were a Family.
3. The Dudleyz: Speaking of Family's, it's not often you can consider a trio of groups that were "brothers". E&C claimed to be brothers, and could've fooled several who weren't wiser about the truth. The Hardys of course are, and the Dudleyz.. well, they claimed to be.. lol
Regardless, to say the Dudleyz aren't one of if not THE best Tag Team, period, would be a mistake. They've won more Tag team Championships than any other team to my knowledge, and it was largely in part due to their time together in E.C.W, but their branching out into the W.W.F.. competing against the two teams above, as well as several others.. that constantly made them a current and common name in the division.
4. The A.P.A: While I wouldn't normally categorize them as an "Elite" team, the fact is during their large period in which they were a Team.. they were dominate and almost undefeatable. The A.P.A were more known for being hired thugs, face or heel, than being a team in it for themselves.
They grabbed hold of the Tag Team Championships, but didn't really seem like a team in too much care or concern to hold the straps, so much as to just be happy in getting into fights defending them. The A.P.A was a top notch Tag Team, because it was a nice team of two powerful individuals, whom were each relatively known for their single's careers. (Ron Simmons prior to A.P.A, and Bradshaw after)
The Up-and-Coming:
5. 2-Cool: While I never thought this team would amount to anything, they did find themselves a Tag Team Championship reign. (or two?) I always seen this as a W.W.F version of W.C.W's "Dancing Fools" (Disco & Wright) but at the same time, unlike W.C.W's version - the fans adapted over time and fell in love with Sexay and 2-Hotty. Why, I'll still never know.
They were a nice team known for chemistry, but not much else. In fact, I only kept them out of my jobbers list because they were Tag Team Champions.
6. Holly Cousins: Often fighting each other as much as they teamed and fought together, Hardcore & Crash Holly were best to watch when at each other's throats.. yet meant to be teaming together.
I never liked watching Crash Holly, but there was just something about him that brought out what I considered one of the best parts of Hardcore's career. The Hollys did become Tag Team Champions, I believe.. and for that reason, the Company must've felt they were something worth giving a chance to.. especially during a time, when the above teams were all present as well.
7. T&A - Test, Albert & Trish Stratus: This is my favorite Tag Team to have never won a Tag Team Championship. I was so majorly disappointed when such a great trio was formed, having seemingly the best all around qualities. A great look, a perfect name (literally), and a nice combination of talent among them.. yet they never won gold. Such a shame to me.
Furthermore, this Tag Team didn't stay together for even a full year, I don't believe. I think they got lost in the fucked up 24/7 Hardcore division, then managed to get lost on miscuing against one another - which lead to Albert turning on Test, Test turning face, and Albert somewhat going on to have a better career. - T&A will still remain one of my favorite Tag Team's to have never won gold.
8. Radicals: I count them, mainly because Perry Saturn, Dean Malenko, Chris Benoit & Eddie Guerrero were all interchangeable individuals that could make an amazing team. Any combination of the four would come to make a great team to face.
In a large part of the time, it was Malenko & Saturn who were often paired together. And as such, they were more thrown to the wolves and jobbed to anyone and everyone. I largely believe this is because the fans at the time (similar to now, actually) weren't fans of solid wrestling knowledge, so much as character, look and personality.. something Saturn & Malenko both really lacked a lot of.
Regardless, I used to enjoy watching both, because they were a great pairing, along with Benoit & Guerrero added into the mix. The Radicals were a foursome that truly rivaled the Horsemen in many ways.. yet never got the chance to prove why, and ultimately were torn apart before they truly got a decent chance.
Jobbers:
9. J.O.B Squad: Al Snow, Hardcore Holly, and a few others collectively made a group known as the J.O.B Squad, and well hey.. they were every bit of what they claimed to be.
Al Snow & Mick Foley largely teamed together and became Tag Team Champions, but it was short lived and often forgotten. Outside of that moment, the J.O.B Squad is largely forgettable - yet still remembered, if that's possible.
10. Boss Man & Bull Buchanan: This is another Tag Team I felt could've had major potential yet fell just short because of various reasons. I think the biggest is because neither individual had much love or care from the fans.
The Boss Man was once upon a time a fan favorite, but that was back during the early 90's. Meanwhile, I don't think "The Bull" ever fully got over w/ the fans. Regardless, I picked this team to round out my top 10 most remembered teams because they had amazing chemistry and a great "story" as to why they were a Team. They just never worked out for whatever the reason may be.