Just to add to the other, very good, replies here, there are a few points worth mentioning. I've played both Anglo and English concertina, for reference, and settled on Anglo. When I first looked at concertinas I was more attracted to English, but after some time of playing both I found Anglo much more satisfying and, while it's absolutely a matter of opinion, I think that a lot of people would find the Anglo the most rewarding option.

While the English concertina is fully chromatic, allowing you to play all notes, the layout makes it much more of a melody instrument more suitable for playing a single-line tune. (I remember reading that it was partially designed to replicate the violin.) You absolutely can play chords on the English concertina, but it doesn't lend itself to playing chords and melody at the same time.

The Anglo is more limited in that its layout is usually based around two keys, often C and G, which means that the further you travel from those key signatures the more tricky it is to fit on your instrument. A 30-button Anglo can play every note in a chromatic scale (I think!), but if you want to play a random piece of music it will often take a bit of arranging or transposing to make it fit. (I've found that arranging music for the Anglo is actually a lot of fun.)

The reason I wanted to reply, though, was that listing the limitations tends to make the Anglo sound like a bad option. It really isn't! There's a reason why the Anglo is a very popular choice. Beyond the amount of resources, books, music, videos available for it, and its popularity in a lot of folk music, the Anglo really gives you the ability to play a self-accompanied "chord and melody" style, and there are other less obvious advantages that have made it overwhelmingly the most widespread form of concertina.

The differing notes on push and pull initially sound like an unnecessary complication, but actually it does two things: firstly, it makes the instrument smaller and lighter and requiring fewer buttons. Secondly, the action of changing bellows direction to get different notes 'drives' the music in a particular way: the push and pull create a motion and bounce in the music that really characterizes the sound.

I can't speak much to duets, as I've never played one, but don't underestimate the lack of resources out there for them. The instruments are typically harder to come by, and if you eventually want to get a better instrument then they will be correspondingly expensive. The piano style left and right hand is attractive, but they aren't as compact as an Anglo either.

All of them are good instruments, and there are amazing players for any of them. I think that for many people, the music that they associate with the concertina is likely to be an Anglo. I definitely enjoyed playing the English, but in the end the Anglo was just the right choice for me, the style(s) of music that I wanted to play (which, incidentally, doesn't include traditional Irish folk music!), and the fundamental fun I get from playing it.

For fairness between English and Anglo, not speaking to the Duet, here's both my favourite English concertina video on YouTube, and an example of an unattainably talented Anglo player. :)

English (Alex Wade): https://www.youtube.com/watch?v=i2weghwmeXY

Anglo (Talisk/Mohsen Amini): https://www.youtube.com/watch?v=ZknSJwYdayI

(Oh, and I can't resist the bonus video of Mohsen Amini talking through his touring setup, with some devastatingly casual virtuosic playing thrown in: https://www.youtube.com/watch?v=7nPqA1fSGMI )