Depends. I have a couple of friends who got their PhDs in applied physics. They do the same thing I used to do with a PhD in Astrophysics. Postdoc doing experimental fusion and plasma physics research. I don't know about undergrad degrees, but with graduate programs, sometimes the 'department' you are in determines less of what you will do for a living than the actual research you (and your advisor) do. I also have friends who were in Engineering Physics at Wisconsin who did/do the exact same stuff, i.e. fusion research.
Ignoring individual factors and history, we are all basically about as competitive for the same jobs as each other. In other words, for a really plum postdoc fellowship at one of the national labs in plasma physics, the fact that you might be Eng. Physics from U Wisc or Astrophysics from Princeton doesn't have as much bearing on your success as does the number and strength of your publications, your individual aptitude, and what the important people in the field think of you. I know this is way more specific than what you were asking for, but it's my experience. Things at the Ph.D. level can be different from industry. Ph.D. level Engineering research shares many similarities with Ph.D. level experimental Physics research, in a way that B.S. level industry Engineering work often does not.
My 2 cents about the undergrad degree and worrying about job prospects and what you want to do. I'm not sure many employers see that much of a difference between applied physics and physics. I wouldn't be surprised if they had two bins, one for engineering and one for all of the majors with the word 'physics' in them. And they many don't hire out of the 'physics' bin. I do not know what they specifically think about Engineering Physics.