(also posted on my Tumblr)
...and it has only a small part to do with who is or isn't the Doctor. I fully admit to be a rather opinionated David Tennant fangirl. I also admit that for the ease of discussion on low text social media sites like Twitter, I'll frequently simplify the issue to missing My Doctor and leave out the greater other issues.
I would say that Series 1-(specials) is a different show than Series 5 and on, but it's more to do with an exchange of almost the total cast, production team, writing staff, and everyone all the way up to the top. Of the non-Doctor characters who are not simply referred to, what we have left is River Song (who did only appear in a single two part episode). But hasn't this happened before? Of course. I never meant to suggest that Doctor Who can be broken up into 1st-10th Doctors and then 11 standing off there by himself. It has changed many times before, and will continue to do so. Like Matt Barber who made an argument that Doctor Who isn't a single series
, I feel it's foolish to just look at the thing as a whole and expect it to make sense. While I don't agree on all of his divisions, the concept stands. This also means that I feel free to judge each portion of it on its own merits. As much as I like most eras, I also wouldn't freely exchange watching one era for another when I have a craving for one in particular.
Like a band who changes enough members, at a certain point it feels like another band. The band can use the same name, but things change. It's just easier to point this out when the front changes than when a manager or the lighting guys change. They're all important, but most aren't the public image of a group.
This of course leads me to discuss the issue I've heard with people saying that people didn't get particularly upset when Christopher Eccleston left. There are honestly a few reasons for this. I know I didn't actually start to watch the new series until after his era. Going in when I knew someone only had one year made me a little reluctant to do any major bonding. It was sad, but bittersweet. He changed, but the rest of the cast, tone, sets, and story moved on otherwise the same. It was noted that he changed, but Rose was still there, in the the TARDIS that remained exactly as it was. It was one guy that changed. Compare this to when Tennant left and we're reminded to dwell on the change as, everyone he knows is left to go on elsewhere and his TARDIS looks drastically by the end of Smith's first full episode. Even when Tom Baker (another long term favourite) left, most of the major productions change happened the series before he regenerated... and his companions were kept along until the next series for the next Doctor.
This is all very rambling, but what I meant to get at is that when a lot of us say it's a different show we don't mean it's strictly because of a switch in Doctor. It was a major change with a lot of people leaving and a lot of new ones coming in all at the same time. When we're lazy, or if we're feeling particularly emotional, it becomes easier to point to the one person who represented it all in our minds... but it's not just about David Tennant. (of all the phrases in here, I expect that one to be contradicted the most)