Well, I only have one, so obviously he's my favorite. But I honestly think he always will be, even when I add another dog. He's mah boy.<33 Like, I don't think I'd feel bad about leaving new puppy at home and taking Jackson, but I probably wouldn't often take new puppy and leave Jackson. Things of that nature.
In regards to being in the house, though, I don't think they'd be treated much differently. I plan on having my future second dog in my bed too (unless it was a larger hairier breed, I MAY make a consideration of not doing it). Both would be allowed on the furniture. Both would probably get to do the same amount of stuff in regards to training, walking, exercising, etc. Those type of things.
With new puppy, I'd also probably get them acquainted with things like doggy daycare, and leaving me more often, and things of the sort. With Jackson, I don't think I'd ever be able to leave him at a daycare or boarding place. He's just not that kind of dog. He's sooo attached to me and I love it but with a new dog, I don't think I'd have an issue bringing new pup to daycare if I go away, and having Jackson stay with my dad per usual. Stuff like that.
So I think Jackson, being the first and all, will always be #1 and hold a special place in my heart, but of course I'd love a second just as much but I think just in a different way.