Seriously, I never understood why Team Rocket wanted Pikachu, there are over 9000 Pikachus on Earth, why that one in particular, what's so special about Ash's Pikachu that distinguises him from other Pikachus, NOTHING, that's what, he's just another Pikachu, nothing special about him at all.....
I think it was in the first ever episode, Team Rocket say that Ash's Pikachu is incredibly strong and different from other Pikachu. They never really elaborate on that in the episodes.. but I think that it's a rather temperamental Pikachu, who, when pushed to the limit at least, gets insane amounts of power, whereas when it's placid and relaxed, it's power level is much lower.
Knowing how Giovanni mistreated Mewtwo, it certainly wouldn't be hard to push this Pikachu to the limit with his skills, and then BAM, harness that power.
Though I'm not sure whether Giovanni actually knows about Ash and Pikachu in detail, I'd assume he knows more about the powerful trainers of Kanto than they know of him.
I think at first the TRio were legitimately like "HEY, WE SHOULD GET THIS AND THE BOSS WILL LOVE US" but as time went on, defeat after defeat, it got to the point where they had to capture Pikachu so they'd feel as if they actually had a purpose again. Where they once found purpose with Biker Gangs, and eventually Team Rocket, in their eyes, somehow the only way to solve all their problems (that they constantly run from) was to capture the "Pikachu on the Pedestal"
I think all Team Rocket needed was some confidence boosting and some self-esteem, myself. They seem to kick ass when they are insanely confident.
In Best Wishes, they've had just that. Giovanni says he has a "special task" for them, sends them to a new region, and they're getting
daily Mission Briefings!
They feel like all their hard work has paid off, that they're finally getting somewhere, so they naturally perform better.
Now, Ash... he hasn't really worked all too hard. He's had it relatively easy. He hasn't come from a crappy family background, hasn't been an outcast, and since early in his journey, he's always had someone to cook for him just like his "Mom" used to. He's had it EASY! So I guess that's why he regularly performs kinda crappily in every region, because he's confident he'll just win because he's supposed to win.