I’m from Mexico and the way I learned the musical notes was by their sound, however some friends from other places learned the notes with letters.
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They aren’t (necessarily) the same thing. The do-re-mi (solfège) system does not (generally) fix “do” to a particular note – it’s just a scale of set intervals (tones and semitones). You could start it at C, or D, or A, or G, or any note you choose. In other words, it depends on the key you’re in. Solfège is useful to do things like sight reading, because if you’ve memorized the do-re-mi scale then you know what different intervals (e.g. thirds, fifths) sound like, which is enough to figure out what a piece of music sounds like from sheet music (if you also know the key).
The letter system, on the other hand, assigns names to particular frequencies of sound. Middle C, for instance, in standard tuning, is a frequency of 261.626 Hz. The C one octave above that is exactly double that frequency, and the C one octave below is exactly half, and so forth. It’s still a system that is relative to a reference, because you have to pick one frequency that you align everything else to. The most common standard is to pick A4 (A above middle C) = 440 Hz. Other standards may put A4 at slightly different frequencies, but it’s not like solfège where you can set ‘do’ to be literally any note and start the scale from there.
Now there is something called “fixed-do” solfège which is used in a number of places (including Mexico), where they decide that do = C. In this case, the do-re-mi naming system is equivalent to the letter-based system, and it’s just a matter of preference (you could equally refer to the notes by colors, for instance – it’s arbitrary). Personally I would say it makes more sense to use separate naming schemes for relative and absolute scales, but there you have it.
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