Two major kinds of difference: mechanics, and style.
Anglo - bisonoric
I.e., each button has different notes when you push and pull the bellows.
English - unisonoric
Each button plays the same note on both push and pull.
Anglo - "spastic"
As you play a scale, you (mostly) reverse the bellows for each successive note.
English - "schizophrenic"
As you play a scale, successive notes alternate between the ends of the instrument.
Anglo - "horizontal"
Hold your hands palm down in front of you. If they were holding the ends in that position, then in going from lower notes to higher ones you would progress roughly from left to right.
English - "vertical"
Similarly, in going from lower notes to higher ones you would progress from nearer your body to further away.
Anglo - regular pattern in two keys
In the two "central" keys (invariably a fifth apart), playing a scale involves successive buttons in a single row (assuming that a "single row" folds over from the one end to the other). The pattern in other keys is quite different, and in fact quite different from one key to another. In fact, on a 20-button anglo, it's only possible to play in two major keys, so a 30-button anglo (which has an additional row of buttons to provide the notes missing on a 20-button) is highly recommended.
English - regular pattern in 8 keys (of the 12 possible)
The pattern consists of 1) an alternation between the ends and 2) on each end an alternation between the two sides of the center line. A minor difference between these keys is that in the pairs of buttons to either side of the center line, each key uses only one button of each pair, but each uses a different set of the outer buttons of the pairs. But there's also a strict pattern to which buttons are outer for a given key. The other four keys have different patterns, but each of those patterns also has its own regularity. (And those are the keys with more flats than 3 or more sharps than 4.)
Anglo - bar and strap
To hold an anglo, your hand slips into the space between a fixed "horizontal" bar (usually wood) and a leather strap. This permits a goodly amount of sideways movement of the fingers, but restricts "forward and back" mobility. That fits the keyboard layout, which extends more "horizontally" than "vertically".
English - loop and plate
To hold an English, the thumb fits into a small loop of leather on one side of the button array and the little finger into the space under a raised plate on the other side of the array. (Some people use the plate passively or not at all. I use it more actively, using the thumb and little finger together to "grip" the end.) This limits sideways mobility, but gives much more freedom for "forward and back" mobility. That fits the keyboard layout, which extends more "vertically" than "horizontally".
Anglo - "bouncy" (or various other similar terms)
The bisonoric nature of the anglo forces bellows reversals at points which may be dictated by the instrument itself. This generates a certain "rhythmic" quality (not necessarily regular; that's still up to the player) which many anglo players prize, at least when playing dance tunes. Others (or even the same ones) work hard to avoid this bounciness when playing more lyrical tunes.
English - smooth
Bellows reversals aren't dictated by the notes you play, and are only forced if you continue without reversing until you reach maximum extension or compression. It's possible to get the same bouncy feel as with the anglo, but for most people it seems to take extra concentration and practice. On the other hand, a legato (smooth) lyrical style comes quite naturally.
Anglo - chords
Because on the anglo the lower notes are in the left hand and the higher ones in the right (with a little bit of overlap), it's possible to play chords in the one hand while playing a melody entirely (or almost entirely) in the other hand.
English - chords
Because on the English the entire range is split evenly between the two hands and most notes are found only in one hand, it is easy to play either chords or melody, but in most cases difficult to impossible to keep a steady rhythm of regular chords going against a melody. However, some other sorts of harmony, including parallel thirds and intermittent or irregular chords, are fairly easy.
Anglo - some chords
On the anglo many notes are found in only one bellows direction. Some chords are quite easy, and others can be difficult, but some are not possible at all, because some of the required notes are available only on the push and others only on the pull.
English - all chords
All notes that are available are available in both directions. This means that any combination of notes can be played simultaneously, limited only by your ability to reach the buttons with your fingers.
Points of view regarding Irish music:
There are persons who claim that the only concertina on which one can make Irish tunes sound "Irish" is the anglo. In my experience, these are all anglo players who concentrate on Irish music, though maybe a few players of bisonoric button accordions, as well. I (you might guess) disagree.
If you want to sound specifically like an Irish anglo player, then the anglo is probably the best way to go. (It's possible to get that sound/feel on an English, but few people do.) But in detail the "sound" or "feel" of a tune played on the anglo is quite different from the same tune played on fiddle, flute, pipes, banjo, etc., yet they're all "Irish". If you don't have any concertina players at your local session, then you probably don't have any anglo chauvinists, and a player of the English won't be criticized as not sounding "Irish" when what's really meant is that they don't sound "anglo". And the English is, after all, as different from the anglo as a mandolin is from a fiddle. (More so, since the fingering of the latter two is essentially the same.)
You also mentioned singing:
The Irish don't have a tradition of using concertina to accompany singing, but elsewhere (especially in England and the US), both anglo and English are used extensively and effectively for that purpose.
Now I see that Catty has offered you the chance to try these two different types. That should be worth much more than all my writings, though I might hope that the above commentary will give you some idea what to look for and try on them both.
Enjoy! (I'm sure you will. )
Man, I hate to ask, but how do duets fit into this complicated picture?