What you're asking is a little difficult to give an exact answer to, as every program is so very different. I did an AEGD at a hospital. GPR's are in general more hospital based, however beyond that, its difficult to group every program. Both are one (sometimes two) years of training past dental school. Most programs have a particular focus or thing they're big in, and other things they are weaker in. For example, with the AEGD program I run, we are big in Endo and Extractions, but we don't do implant placement. Other programs do a ton of pros and implant placements, but you're only seeing 2-3 patients a day, and so you don't get your speed and efficiency up. Some programs have you doing a ton of specialty stuff, but you're still in a dental school environment. Other programs may get you hospital and special needs experience, but might be more limited in your standard day to day dentistry than other programs.
AEGD and GPR's are becoming much more popular than they were 20 years ago. About 80% of my class went into residency of some type, mostly AEGD/GPR when I graduated in 2013. What has become a very true and sad fact is that many dental students aren't getting the same experience in dental school that they did in previous generations of dentist. My faculty had to do 30+ crowns to graduate, I had to do 8-12 crowns, I've talked to dentists who graduated recently who only did 3 crowns in dental school. I've spoken to people from big name schools who did their first Class II cavity prep on a real person on their board exam. It is very common for me to hear at interviews that someone has done less than 50 direct restorations to graduate, or less than 6 root canals - all from big name schools.
Now, this isn't every school, but unfortunately many are producing graduates who don't get the experience they should have had. Also, every student from every schools gets a difference experience as well.
What you need to consider when you're thinking about doing an AEGD or GPR is if you currently feel ready to go out on your own and at least have the whole range of tool box of skills to manage most situations. It's not to say you can't get experience if you just go out in the world and start working - you will, and you'll pick up new things, but it's just very different doing CE classes vs. having someone walk through things with you step by step the first time or two. Some people want that, some people don't - it's really up to you.
What my mentor told me at my interview for my AEGD that sealed the deal for me was this: Yes, you'll be making a lot less this year than if you were working, but you need to think of it as taking a year and investing it in yourself. That investment can often pay off in big dividends if you're comfortable with a wider range or procedures, or you're able to keep more procedures in house.
Moving on to how to pick the right AEGD or GPR - as you are entering 4th year, you need to start thinking about what areas of dentistry you feel comfortable in, and which you don't. Which areas of dentistry you like, and which you don't like. And for those areas you don't like - is it because you really haven't had much exposure to those yet, or had a bad experience or two? What kind of procedures do you want to get more experience in?
Once you've done that, you need to start researching programs. SDN here is a great resource. Most times most AEGD/GPR program websites are terrible and don't really tell you what you need to know. Feel free to email programs and ask to talk to current residents about what they do. Go visit some programs in geographic areas you are interested in.
You want to find a program that is a good match for you - not just the clinical strengths, but also the clinic, the faculty, the environment.
On your question about how "competitive" programs are - in general AEGD/GPR are less "competitive" than other specialty residency programs. I'd say the most competitive General Dentistry programs are the VA GPR's. These are great programs that can give you a really wide range of experiences in many areas, and you can work with great specialists. These programs get tons and tons of applicants, making things more "competitive." Beyond those, I think things are very different. For my program, as long as a candidate is above a certain bar, grades don't mean much to me. A 3.25 isn't very different than a 3.55 to me. Though if you got a C in your FPD class I'm going to ask you about it. I'm not looking at just grades, I'm looking at personality, drive, passion - things that don't show up on a transcript. I've rejected candidates with a rank in the top 20% of their class if they just don't have what I'm looking for, or if I don't feel like my program would be a good fit for them. I'd like to think that many program directors are of a similar mind, but who knows, maybe I'm the crazy one!
There are a ton of programs out there, and many don't fill their spots. My first time around I ended up doing post match (I made the mistake of only applying to 2-3 programs and lost out on the MATCH game) - I had a fairly "big name" program from a major city call me - one that I didn't even apply to, but they found my application on a list, and offered me a position sight unseen after a 10-15 minute phone call. So in conclusion about the "competitive" nature of AEGD/GPR, there is a very wide range, and so it is very difficult to say exactly.
And to your other point, you're totally right, these programs don't make you a "specialist" but there are many nice aspects of being a "generalist." 😉
It's a tough decision to make, but I can honestly say that my doing an AEGD was the best possible decision for my career, and it helped turn me into the much more confident and capable dentist I am today. That said, that was what was right for me, and that might not be what was right for you. Good luck!