If there's any difference between the remasters and the originals, it's probably because of hardware issues. Especially on Android, all the various devices have tiny little variations in how they deal with inputs (be it touch inputs or physical presses). I've played the games on the Xperia Play (which I understand is one of the devices Sonic CD remastered was built for) and they seem identical to the originals. (It's also worth noting that the Xperia Play is one of the few devices from that area that has a gamepad built in, so it doesn't have all the issues that come with bluetooth controllers). However, I grew up with the PAL releases which I believe were a tiny bit different from the NTSC versions. Although again, I am bit unclear how all this works and so might have misunderstood.
It wasn't that straightforward. Originally Sega just wanted to release emulated Mega Drive games on iPhone. Sega of America asked on their blog for people to list their favourite titles. This was around the time that Christian had shown off his Sonic CD on iphone protoype (with the lovely widescreen). That became the most requested title on the blog, so Sega of America reached out to Christian. It's not clear if they gave him source code access after hiring him. Sonic Team did closely supervise the title and were adamant about keeping it close to the original (which is why Christian's plans for new levels and a final boss were thrown out). For Hidden Palace in Sonic 2, Naka himself had to play and approve the level. Given Sonic Team's mandates, I'd be surprised if they didn't give the source code, but this has never been confirmed.So Sega greenlit remasters based on fan reverese-engineered code rather than supplying the originals. That does seem characteristic of them, honestly.
Another issue is that the funding for the projects was a little messy. Sega of America seems to be driving force behind the projects, but it was paid for by Sega Networks (which was Sega of Japan's separate mobile company). So it's not clear if that would have influenced what Christian would have had access to.
There was an interview a while back with M2 who were talking about the difficulties of adding a Spindash to Sonic 1 for the Nintendo version. From their comments, it's sounds like although they had access to Naka to ask him for advice, they had to recreate a lot of things to get the Spindash working, which suggests that maybe they didn't have the source code.