In general, chords with the notes closer together are called closed voicings, while chords with the notes farther apart are called open voicings. For simple triad chords like the ones you mentioned, a chord is closed if it occurs within a single octave and the chord is open if it is spread out across more than one octave. More generally though, a chord is closed if it's the most compact voicing of the chord.
There is another technique worth mentioning: the example you've described is called a "drop 2" voicing or drop 2 chord. This is when you take the second highest note from a closed voicing and move it down an octave. The other notes remain unchanged. In your example, you started with a 2nd inversion C Maj triad (G3-C4-E4). Then we take the second highest note (C4) and move it down an octave (to C3). This gives the final drop 2 voicing of C3-G3-E4.
Here's a more elaborate example. The bottom shows the exact same voicings, but with the second highest note dropped down an octave for each chord.
If you are working with 4-note chords, you still take the second highest note and drop it down an octave. For example, here are some repeating Dmin6 A7♭9 chords. The bottom shows the same chord voicings, but with the second highest note dropped down an octave.
This is done to achieve a more open/spread out sound. Being able to play both increases one's musical vocabulary. That's generally helpful, because variety is a lot of what makes music sound good. Playing just closed voicings for an entire song can become boring for a listener. Using drop 2 voicings adds variety.