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Please take a look at this arrangement:
Why does the arranger/composer play
while he could just play
They are the same notes, they sound the same and are easier to play opened. What advantages brings it playing them fretted? Can someone tell the intention?
Two big factors affect the decision to play open versus fretted:
Sound: Open notes sound brighter and tend to sustain longer. They are great to use for pedal tones for this reason. But that means they can sound too loud or otherwise stand out next to other fretted notes.
Playability: In certain cases, open notes are actually harder to play than fretted notes.
Specific to the piece in question:
In the last quoted measure, you couldn't play both the high E and the high F# (second fret on the E string) at the same time, or let them ring out together, so you have to fret that E. Also not shown but implied is that in that last bar you would barre the second fret with your first finger to take care of the A string and high E string. Once you barre, you can't play open any more.
Taken all together, the E having to be fretted in the last measure and the elimination of string skipping in the first measure are enough to justify playing the notes that don't change through all four measures fretted, so you don't have to move your hand just to play the same notes. Also it will sound different if you don't move your hand since those two notes can just ring through all four measures. I would absolutely play this passage as notated if I were playing it, and I routinely change how things are played to fit my style.
Which brings up an important large point: Try it both ways for yourself and play it whichever way works better for you. I think I've explained why fretting the notes can often be a better choice, but in the end it's going to be you playing it so you decide how you play it.
Here are two factors to consider, which would determine why a player would decide on an open string, v/s a fretted note.
Convenience - If the passage you are playing is fast, you won't have time to move to a less convenient part of the neck. If speed is not an issue - it could be that it's simpler to allow the note to ring open than to move your hand.
Tone - Each note played in different positions has a different timbral quality. Listen to the same note played at every possible position on the neck. They all sound unique. An open string can also ring, while a fretted note will only sustain as long your finger is applied.
Adding to Tim's answer below: playing an open string with vibrato can be achieved by moving the hand you use to fret the note behind the nut at the top of the neck. Jimmy Page uses this technique when playing Black Mountain Side in a live Led Zeppelin video. Pretty cool technique - but more effective for showmanship than practicality.
Adding to Todd's answer - fretted notes can always have vibrato applied, to make them usually sound better - not easy on open string notes! However, here, with a beginner piece, it's probably not the reason. I thought maybe to play consecutive strings, but the subsequent bars put paid to that theory.
I think it is done so that each note can ring out as long as possible. To let all the notes ring out in the final bar requures the given fingering to give one note per string. Changing the hand position for the last bar would kill all the currently sounding notes and create a noticeable glitch As it is you can plant two fingers on the 2nd and 3rd string and leave them there for the duration.
