Giving Doctor Who to BBC studios is almost certainly a cancellation and potentially show altering (to chase better ratings) time bomb and I don't like it, to be honest. Being funded by taxpayer money limits the show in budget, but it also has allowed it to survive and do things where, in any other circumstance, it probably wouldn't. It'll lead to a harder push for films, adjustment of the show's style, and so on. Maybe not immediately, but eventually, I'm confident. So it makes me nervous. I don't want Who to end up like Star Trek, the second longest tv sci-fi show after it.
That said, BBC Studios is an interesting one as while profit driven it still serves the public interest side, so not sure how it could/will work. It's a half-step, in a sense, but that could lead to a full step up later on, bringing Who outside of the BBC's regular sphere of influence. At that point the show would be in the same place it was in around 1996, where Fox owned an interest and the BBC owned the rest, and the result was no series after the McGann 'movie' pilot and years of rights wrangling (that's still going on - that's why no movie characters have ever appeared again, not even in Big Finish).
It's worked for Top Gear because people like Clarkson and Hamilton head up its production company, and they're truly invested in the show. The thing is, many of the things TG has done, Who has done anyway through partnerships forged through BBC Worldwide while keeping the show itself ringfenced away from it. This is things like the orchestra concerts (or Top Gear Live), the Experience, etc. The moment Doctor Who is 'just another show', it's in trouble.
What could change in real terms? Being outside the actual BBC TV ecosystem will adjust a few things. For instance, it makes it harder for last-minute deals to happen that have resulted in things like extended run-times, or the Doctor/TARDIS appearing on BBC One Idents, etc. These are good examples; in the current setup, everybody is on the same team, so when RTD went begging for an Ident at Christmas, the BBC not only said "sure," but provided the budget to do CGI of Reindeer dragging the TARDIS around and stuff. It's in their interest. Once the show is on a different 'team', they have to be more careful about showing bias and all that crap, and wham, stuff like that gets harder. Julie Gardner ringing up and saying "We're ten minutes over, are we okay?" changes from "we'll make it work," to "no, cut scenes," more likely.
It's little things like this. For what'd give us some better CGI, it's not a good trade-off.
Another big example relates to Wales/Cardiff -- the show has been made there since 2005 and basically exists because the BBC made a commitment to the Government to do what they could (as essentially a government arm) to increase industry and provide jobs in Wales. It's difficult to quantify how much this meant to Doctor Who, but it's a lot - RTD was approached and given carte blanche to make what he wanted essentially because he was the highest profile Welsh TV writer and this was all about developing big, ballsy new products in Wales, since the Government had asked for it. Shows like Doctor Who, Casualty and Sherlock have been filmed there and created hundreds if not thousands of jobs, and prompting the building of the new Upper Boat building and all that. When Doctor Who was having an 'off year' for the Davies/Moffat handover, things like Sarah Jane, Torchwood, Merlin and so on were made to fill the gap and ensure people weren't let go. This is part of what the BBC does outside of just 'making TV' - they've built an infrastructure. Obviously the non public service parts of the BBC don't think in this way and follow a much more typical hire-then-dump up-and-down staff/infrastructure build like the rest of the industry, because they're built around profit... so another worry is a shift like that could mean a lot of the talented people at Upper Boat lose stability, which in turn might mean they go elsewhere.
It also would mean things for international licensing - Worldwide would have to bid a lot more I imagine and wouldn't get any preferential treatment, so in the US it could end up elsewhere if somebody wanted to bid in whenever the contract expires. Lots of little considerations like this.
Also: I always think the budgetary constraints define the show, really. RTD/Moffat/Tennant once had a great conversation on a commentary about how the bedrock of Doctor Who is how "The scene of 50 million Daleks descending from the sky" is followed by "a scene in somebody's bedroom." Tennant says the show is built around a lot of "standing around urgently talking" -- and this is true even in the big budget episodes like Day of the Doctor. It's what sets it apart from other properties, IMO, and more money is dangerous 'cause of it.