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For example, in a C Major chord of (C, E, G), I have the intervals of:
In a C Minor chord (C, Eb, G), I have intervals of:
So I have the exact same intervals, but it sounds very different. Is this because the intervals involving the root note carry more importance, or does it have nothing to do with the intervals?
The intervals in a chord do define what the chord sounds like and how it functions or operates, but the intervals are felt in relation to a root note. (And as explained later, in relation to a tonic) Where does the root come from? Perfect fifths and fourths have a special role in pointing to a possible root. In addition to that, whatever is the lowest note i.e. bass, is a dimension in the chord, but it's not the same thing as root. G - C - E does work as a C major chord, but if you're expecting C as the tonic, home note, then G - C - E doesn't feel as much like being home as C - E - G.
The C - G interval in both of your example chords is a perfect fifth (or fourth if it's G - C) point to C being a possible root, and the "third", E or Eb, is felt in relation to that. It doesn't matter which of the notes is higher, C or G, the interval points to C being a root. G - C - E is a C major chord, because G - C points to C being a root. But if we move C down a half step: G - B - E, then the root-defining fifth/fourth is B - E, which points to E being a root, and the G makes it an E minor.
If there are no fifths or fourths, then the bass note has more weight in defining the root. C - D - E would be some kind of a C major chord, but C - D - Eb is some kind of a C minor chord. But E - C - D? At least it's not E minor or E major.
One more thing that affects how notes in chords are perceived is the tonic, the expected home base note. For example if you heard your example chords after having established Bb as the tonic, it would make you feel different about them compared to if, say, C was the tonic.
It's possible to have several fourths or fifths in a chord, and then the effect or "meaning" of the chord comes even stronger from your expectations. What is a G - C - F chord, is it a Csus4 waiting for the F to resolve down to E, or G7sus4 without a fifth waiting for the C to resolve down to B? Or maybe "F sus2", not waiting for any resolution that much? What it feels like depends on what you expect. Even rhythmic placement in a phrase or other longer structure has a role in setting expectations. You can experiment with this by establishing a tonic by playing a cadence. For example, establish F major by playing the chords F, Bb, C7, F. What does a G - C - F chord feel like? Then establish C major: C, F, G7, C. What does the G - C - F chord feel like now?
To put it all together, the sound and function of a chord comes from: all its intervals, in relation to (1) a root note, (2) the bass note, and finally the/a (3) tonic note which gives an additional point of reference. You could even add other note-expectations as points of reference as well. Like for example if you expected C - Eb - G, but got C - E - G instead, then it feels a lot different than if you had expected the C - E - G.
There are non-functional (if that's a music term) aspects to the sound of chords as well, namely chord voicings, which means the spreading and/or doubling of notes in different octaves. A wider spacing gives a different sound than a narrow one. If you move your E or Eb up an octave, so it's C - G - E or C - G - Eb, it's a different sound even though the major/minor aspect doesn't change.
Edit. The OP asks:
Just to clarify, what's the difference between the tonic and the root?
A chord has a root note, but a tonic exists outside individual chords. There's no tonic in any single chord, but there can be a tonic in your mind. When people say that a song is "in C", it means that C is the tonic. It is a home note, center of balance to which your mind relates other notes and chords. When analyzing or listening to a single individual chord, naming a tonic would be conceptually incorrect, because tonic is not a property of a chord.
When exactly is a tonic set? It's a slightly fuzzy phenomenon that needs time to settle in, so it's not an immediate on/off switch. If you've spent awhile without hearing any music, then hearing even a single chord might set the tonic. But if you had been listening to a song that's in F, hearing one C major chord immediately after that most probably won't set the tonic to C. You will hear the C major in relation to the F and it won't feel like home. In this case, resetting the tonic somewhere else can require more powerful means. Keep a moment of silence and play a cadence.
If you play the chords C, F, G7, C, it will most probably cause C to be established as a tonic note in your mind, and C major will feel like the tonic chord. If not, play the cadence C, F, G7, C a few more times. Now, having C as an expected home, C major will feel a certain way, and F major will feel a certain different way, even though they are both simple major chords.
These feelings would be different, if you had a different tonic in mind. Try it. Establish G major as the tonic by playing G, C, D7, G a few times. How do the C major and F major feel now? C doesn't feel like a home anymore, and F feels quite different and special, doesn't it? Even a simple chord can sound unfamiliar and more complex, if it's played in an unexpected place.
You don't have the exact same intervals... above the bass.
You don't want to just list the intervals and then mix up the order and say it's all the same.
Traditional harmony puts the intervals in relation to the bass (or you could say in relation to the chord root, it's not the same but we can skip over that.)
C major is a major third above the bass and a perfect fifth above the bass.
C minor is a minor third above the bass and a perfect fifth above the bass.
The interval qualities do determine the sound:
Sounds are perceived according to their relationships with other sounds — so ... intervals. And intervals are perceived — and defined — according to the lower pitch.
Given the example of major and minor chords, while they contain the same (unordered) collection of intervals, the order is important, and it's clear that the defining relationship in terms of chord quality (major vs. minor) is between the third and the root — when the chord is in root position.
This is borne out by the fact that a common difficulty in ear training is distinguishing the quality of a first inversion chord. First inversion major chords are often misinterpreted as minor, and minor chords as major, because of the interval against the lowest pitch.
So, the root of the chord is critical in defining its sound, but experience suggests that the lowest pitch is even more important until one is trained to perceive the root regardless of its position in the chord.1
Historical aside
It's worth noting that there is significant historical precedent for defining harmony in terms of the lowest pitch. Prior to and concurrent with the development of Tonality, figured bass notation, for example, defined harmony in terms of intervals above the bass. Further, Rameau's theory of harmony based on thirds, which is now the predominant understanding of chords, stemmed from the idea of a "fundamental bass" -- an idea that became understood as the "tonic".
Note
1 As @piiperiReinstateMonica points out in the comments as well as an answer to the OP, the location of the root of the chord is most clearly signaled by the presence of the fifth (in root position) or fourth (in first or second inversion). In ear-training courses, learning to find this interval within the chord, and to identify whether the lower or upper note is the root (i.e., whether it's a fifth or fourth, respectively), is essential to overcoming the initial difficulty created by hearing, for example, a minor third at the bottom of a major triad (i.e., a first inversion triad).
