[gjm11]

The UI, the user interface, is the part of a system that you directly interact with. The UX, the user experience, is what it's actually like to use it, which is affected not only by the UI but also by other parts of the software.

"When I press this button, nothing happens for five seconds and then it displays the results." Bad UX. Probably partly the UI (if something might take a while, the UI should at least show some sign that something is happening, preferably in a way that gives some clue how long you're likely to be waiting). Probably partly not the UI (maybe the reason it's five seconds rather than 0.1 seconds is that the internals of the program are doing something time-consuming; maybe that's unavoidable, maybe a better algorithm would fix it. Or maybe it's waiting for a server somewhere else, and maybe smarter cacheing would help. All of that is UX, none of it is UI.)

"When I open the app, the first thing I see every time is an annoying popup inviting me to buy the premium version. I have to close it every time." Bad UX, at least from the user's point of view (it might be exactly the experience the business folks want users to have). I suppose the popup is part of the user interface, but the actual problem here is a business decision (advertise at users in the hope that they will be convinced and/or pay just to make the advertisement go away) that's contrary to users' interests. The best UI designers and implementers in the world won't change this.

"This thing is incredibly confusing to use". Bad UX. Might be the result of bad UI (e.g., no one has thought through how the menus should be organized). Might be because the underlying software is designed in a way that doesn't fit this user's brain, and changing this would need deep changes to the insides and not just UI changes.