Why do we base a barre chord on two different shapes ? Is the E shape more important than the A shape ? Why can’t I use the E shape all the time ?
Use the E shape barre all the time if you want. It's not a problem. Until about the fourth or fifth song in, when it starts to sound boring, because it's so samey.
Your premise that barre chords are based on two shapes is flawed. They're based on many shapes, and for me at least, the 'caged' system makes sense in that respect. Reasoning is that each of the open chord shapes, C, A, G, E and D can be transferred to anywhere else up the neck, by using a barre behind, just as the nut was. So, for example, in your case, an E shape on 3rd fret barre produces a G chord. However, using the C shape, barred at the 7th fret give G; the G shape, barred at the 12th fret gives a G; and a 4/5 string barre at the 5th fret gives a G when you use a D shape. Not forgetting 10th fret barre A shape for yet another G.
With all this choice of different voicings, the sounds made are more interesting, at least.
Another problem with only using an E shape is that there's a lot of movement required at chord change times. This makes the rhythm sound amateur, with, in a lot of (self-taught) guitarists resorting to strumming a lump of open strings while they move positions. Unnecessary, and not sounding good.
Let's take a simple A/D/E song. The A could be on 5th fret, E shape, D on 5th fret, A shape, but instead of sliding that up two for the E, barre on 4th, use C shape. Or even lazier - play an open C7 shape on 5/6/7 frets. Don't knock it till you try it a good few times!
Having said all that, the E shape may be construed as more important by some, on the basis that it's a full 6 string chord in root position, thought often as the most stable. Whereas the A shape can be played with all 6 strings, giving the 2nd inversion, but often advocated to be played without the 6th string. So, only 5 strings as opposed to 6.
As Tim mentioned in his excellent answer there are actually 4 commonly used shapes of Major Barre Chords not just the two you mentioned in your question. In addition to the E shaped Barre Chord and A shaped Barre Chord there is also the C shaped and G shaped Barre chord.
Basically any chord played as an open chord can be played as a barre chord if you think of your barre finger as a capo.
The E shaped Major Barre Chord and A shaped Major Barre Chord are the most commonly used because they are the easiest to play.
I found the E shaped Barre Chord the easiest to play but quickly learned its limitations when I decided to learn to play Takin Care of Business by Bachman Turner Overdrive. The chords are C Bb and F - all Major but it sounds more authentic to to mix in a 6th with each chord in your strum pattern.
To play the song using only the E shaped Barre Chord I had to cover from the 8th fret to the 1st fret with my fretting hand. But by learning the A shaped Barre Chord I only had to cover from the 3rd to first fret and both the Bb and F major were on the 1st fret. Plus it made adding the 6th easier.
I believe you too will find certain chord progressions easier to play if you learn the A shaped Major Barre Chord in addition to the E shape. For alternative ways to play the A shaped Barre Chord read this Different Ways to play the A shaped Barre Chord
For those who may be wondering why there are not 7 shapes for Major Barre Chords let me explain.
Major chords that can be played on a 6 string guitar in "open position" (meaning some open strings are used as part of the chord) are A, C, D, E, and G (all major). Of course some of these chords may also be played as minor or 7th chords in open position as well.
B Major is not typically played with open strings and the first position of a B major chord is actually an A shaped Barre Chord. Similarly the F Major chord is not played with open strings and is actually an E shaped Barre Chord. Note that an easier version of both the 1st position B and F Barre Chords can be played using only the top four (highest) strings.
Thus the only unique shapes for Major Barre Chords that can be moved up and down the neck are A, C, D, E and G or if you prefer a mnemonic device you can remember these by changing the order to make the word CAGED.
The open D Major shape can be played as a "D shaped" Barre Chord or can appear as a fragment of a C shaped Barre Chord. So if you can't stretch across four frets to play the full C shape Barre Chord you can play just the top four (highest) strings and think of it as either a part of a C shaped Barre Chord or as a different version of a D shaped Barre chord. As you can see in the picture below, the D shape appears in both a "D shaped" Barre Chord and a "C shaped" Barre Chord.
The C, G and D shapes are not as popular for movable Barre Chords as the E and A shapes because they require a stretch across four frets (versus 3 for the E and A shape) between your barre finger and the other fretted notes in the chord.
In summary, I would encourage you to incorporate the A shape Barre Chord into your arsenal as you advance in your guitar skills. I believe you will find that it gives you more options to make certain chord progressions both easier and more interesting. The link provided above may help you find the method of playing the A shaped Barre chord that works best for you.
Good luck and keep learning and expanding. Improving your guitar playing is a lifelong process. Enjoy the journey!
To me, the E and A shapes are fundamentally the same shape, just moved across one string - they have a very similar physical pattern, and the same pattern of intervals. The only reason that the physical fingering pattern is slightly different is that is the interval between the G and B string is only a major third.
One reason not to play chords too high on the fretboard, as you might end up doing if you only know the E shape, is that the higher you fret a string, the less 'ideal' the harmonic series it produces (because as you fret higher, it gets thicker in comparison to its length). So if you play D# with the E shape barred at the 11th fret, you may get a less 'pure' sound than A shape barred at the 6th fret.
On many guitars, it's also quite difficult to play barre chords high up the neck - the action is likely to be higher towards the 12th fret, and on some instruments (such as classical guitars), the lack of a cutaway may be a hindrance.
Once your fingers get used to making the shapes, it may also be faster and less tiring to move your fingers across one string to move up an interval of a fourth than to move your wrist 5 frets down the fretboard.
In summary - in some circumstances it may be easier, and sound better, to play using the A shape.