In terms of the campaign, It was a lot more serious, the voice acting was better, and the dialogue actually sounded *so* much more intelligent and well written than in SC2. Just think of Mengsk's speech...I don't remember anything like that in SC2. The story was also way cooler when the overmind was still alive and he had these beautiful ambitions for the perfection of his race. It was a little sad what they did in BW, but beyond that into SC2 I felt like the game lacked a deeper purpose.
As for multiplayer, you can objectively see that cheesey play is rewarded a lot more in SC2 than in SC1. SoS won the world SC2 championships, with like 3/7 games using cheese in two minutes. This is not possible in BW and requires a lot more strategic thinking.
In general what I found disappointing about SC2 is that it almost always devolves into death balls, and occasionally (with PvP) doing coin-flip attacks and two-base allins that just crush the opponent or leave you in a losing position. I don't remember seeing anything like that in BW. You could lose an army but not lose the game, it was continuous warfare across many fronts that had some 'elasticity' in terms of whether you won or lost.
Plus I would say the multiplayer units are not as interesting as they were when BW was first introduced (i.e. seeing defiler, queen, reaver for the first time? Exciting! But now?? Its just a colossus with lasers or a stalker that blinks. Not so great). They mostly recycled them and repackaged some others, introducing small new abilities but nothing really revolutionary. Maybe its to be expected since it is SC2 but I was hoping for some real evolution and a bit of revolution in the races, but they played it very safe.
On the whole I feel like it was a well-polished game that was fun to play but it had a lot of problems that detracted from its appeal compared to BW. But mostly to me its about the story, and I thought the original SCI cinematics and dialogue were brilliantly done and I missed that (SC2 still good though!)