Milcho made some great points, and I'm going to comment with some of my own perspectives of some of them.
Obviously one cost is time. Lots of loading-screen-like transitions will force the player to spend a lot of time waiting around doing nothing, and probably getting bored. I can't tell you how annoyed I've been when I've had to exit/enter a building in a city in Skyrim twice in a row, spending nearly five minutes just because I forgot to leave some random item. Something that should have taken me a minute tops, suddenly took about five times longer.
I think as long as you make the areas large enough, minimize the loading times
This problem really depends on the scope of your game and the size of the areas. Gigantic 3D Monsters of games like Skyrim and the like are practically forced to have these transitions for the sake of only being able to have so many assets in memory at once. Another issue that wasn't really touched on here was that of the change of scale that occasionally happens between transitions, and those being the reasons for them. This is especially common in 2D games, but I'm sure it's done elsewhere too. How many times have you entered a house only for it to be this sprawling mansion once inside? Not having loading screens (unless you use some really fancy tricks ala something I'm sure Antechamber does, which would be disorienting anyways) means that your assets have to be to scale for whatever you want to be inside of those buildings / structures / caves / whatever.
In a smaller game, loading times are basically negligible (or should be anyways), and shouldn't negatively impact the user's experience. Unless an area is very large in your game, you are quickly moving between non-contiguous spaces, or there is a change of scale (town to world map in old RPG's, things like that) user experience could probably be improved by a fancy camera switch and streaming content in to the world. I know games like some of the later Tony Hawks did this (albeit poorly from what I understand) where you skated though long tunnels that connected their zones. Bar the player cheating and giving themselves super speed, (Conjecture) the new area would have time to load and unload old assets as they moved through the tunnel.
I've also disliked how some games allow you to be in a position to affect another area (say, on top of a wall above a battlefield), but you couldn't affect the other area simply because it was 'another area' and you had to go load into it before you could affect it.
This is huge, and should be avoided as much as possible in your game. If someone is forced to use frequent transitions for whatever reasons, I would much prefer to have awkward loading transitions between seemingly connected areas than being barred off because my skill / ability / whatever can't be used at range because it's at the border of a transition. (and makes sense to use it there.) Not only is it immersion breaking, but it's frustrating, more so than sitting around and doing nothing slightly more often in my opinion. Granted, no one likes loading times, but I don't want them to get in the way of my experience as I'm actually playing the game either. For example, some of the Zelda games do this poorly at times. (I'm talking older map tile based ones). It was more frustrating than anything to exit screen left, only to be faced with an immediate, obvious, 1x1 tile of space that you can walk around in that they decided to put on the other square. This was frustrating for two reasons! One. Now I have to wait (Even if it's super fast) and go back to where I was so I can find a way around, rather than just seeing it clear as day in the last 'room'. Two Now that map square is marked as visited as well! I didn't go there, I simply got a good view and went home. Granted, that's more of a level design flaw in my opinion that doesn't fit well with the constraints of the engine. If you have ideas for your levels, and are building the level loader / transitions around this world, take such things in to account!