This is a very common question, and I've done TONS of remodel projects. Let's analyse it. Your existing home is livable, but there's a few things you'd like to add or change. Let's keep the example simple: you want to move a doorway 3' to the left.
Relocating an Existing Door: (if you don't want to read all of this, simply compare the length of this paragraph to the next one, and you can infer the answer:) To relocate the door, you need to remove the existing door and carefully remove the trim if you want to re-use it. Not hard, but it takes a couple hours or more to pull the nails out of the trim and scrape off the caulk and glue. It's probably cheaper just to buy new trim. Next, you would have to carefully cut a hole in the drywall (on both sides of the wall to access the area where the new door will be placed. You need to be careful not to remove too much drywall, or you would have to replace more wall than you bargained for. Once the drywall is surgically removed, you'll see the structure: likely 2 studs at the top, a stud on the floor, and vertical studs every 16". These will need to be carefully removed, and a new door frame and header will need to be built. (If it's a load-bearing wall, the area above will need to be temporarily supported while you have the studs removed.) Also, there is likely a light switch next to your door, so that will need to be relocated. This requires removal of the old switch, adding a junction box to splice the old wires, installing a new switch, and running the wiring to it while ensuring other circuits tied into this junction box are reconnected. (Let's hope it's not old-school electrical work that requires additional replacement for safety purposes.) OK, now you have a door frame and electrical. In some cases, there may be plumbing or duct work in the way that may need to be re-routed adding another layer of complexity. (Let's also hope you don't find rot inside the wall that would also require additional reconstruction.) We can now cover all of the exposed framing with drywall (on both sides of the wall). Once the drywall is screwed in place, it needs to be taped and mudded on its joints and screw holes. The mud is a plaster that will enable the seams to be covered, so they cannot be seen. Once the mud dries, it can be sanded smooth. This will be easy if the existing wall is "straight and true." It will be difficult if the existing wall has a texture that needs to be matched, or if it was built with the out-dated lathe and plaster approach which is quite brittle and damages easily. Now that we have smooth wall board, it's time to install the door and trim. A standard pre-hung door will slide right into the opening, and with the use of shims (little scrap pieces), you can fit the new door within the "rough opening" of the wall. Some existing walls are straighter than others, so this may or may not go smoothly. Once the door is set in place, the next step is to apply the trim (on both sides). It will need to be glued and nailed on with a finish nailer. The baseboard (and maybe the crown molding) will also need to be extended or modified to integrate with the door trim. Once all the trim is attached, the nail holes will need to be filled with putty, and any gaps will need filled with painters' caulk. Now it's time to paint. The goal is to match the new wall and new trim with the old wall and old trim. You will likely need to paint the whole wall (on both sides). Once the door is painted, it can also be hung ... and you're done.
Building a New Doorway: In a new construction project, there's no demolition, so there's no surgical work or protection of flooring, ceilings, walls, etc. Framing a new wall with studs is easy because the carpenter has total control over the selection of straight lumber and does not need to integrate his or her work with any other existing (faulty) conditions. Pulling electrical wire in a new wall is also easy because there are no other existing conditions to deal with, and the maze of electrical is all exposed and easy to figure out. Drywall will go smoothly because it does not need to match anything else, and the new stud wall will likely be built straight and true. A new door wouldn't have any plumbing or ducts in the way either. Installing the door will go smoothly since the wall is straight, and trim can be expedited because the same conditions yield the same results when there are many of the same door in a new construction project. Working in bulk is quick. Painting will also be a breeze since the whole area can be sprayed without masking off existing conditions.
The Answer: Building new is cheaper. Remodeling has all the same expenses as new construction, but you still need to do additional tasks like, demolition, retrofit, and protect existing areas. Existing construction also has a lot of unknowns hidden under your walls: poor construction, asbestos, lead, rot, rodents, earthquake updates, etc. These problems tend to expand the scope of a project into all of the surrounding areas too. As an experienced architect, I think it is important to communicate all of these likelihoods up front. It is smart to prepare for them and hope they never happen than to be blind-sided with unforeseen circumstances. After seeing so many remodeling projects over the years, I have a good sense for problem areas and can offer alternative solutions to prevent them whenever possible. This door example is a simple one ... imagine a much larger remodel project.