The great divide between classical and other styles of music has been so intense at times that it has resembled modern day American politics. Classical was “right” and everything else was “wrong”. Not many singers of other styles also shared the opinion of diametrically opposed sides but some did.
My wife is a professional dancer, so looking at another art from could shed some light. In ballet training, there are things which greatly improve specific things with a dancer. A ballet-trained dancer may have a better line, will dance with shoulders down instead of raised, will have excellent balance, a sense of center, may have better turns, better extension, may have coordination and better communication between mind and body and may also be able to learn steps and routines faster, easier and better. All this comes with a caveat. If the style of ballet is carried over to hip hop, to jazz, to tap or other styles, it will make the other styles look stiff, too smooth, or even silly. With so many dance competition shows on TV, you may have seen this.
The reason I gave the example of dance is that it also applies to singing. I know of one or two classically trained singers who can sing Pop and R&B and without a hint of anything classical or operatic. Julia Migenes and Josh Groban can do this. Many classical trained singers either cannot or will not.
You must have heard the difference in how a classical singer sings, as compared to a singer of popular music. Proper training will result in breathing correctly, no register transition area issues (no cracks), having a fully developed and extensive range, having control over all dynamics, and having excellent endurance.
Some differences between classical and other styles are: tone, timbre, articulation, rhythmic structure and its interpretation, pronunciation, enunciation, vowel formation, and musicianship. This is not to say that a classical singer has poor musicianship but a jazz singer may have musicianship which extends beyond the norm of a classical singer, particularly with modes, extended chords, altered chords, and other harmonic construction not found in classical music. The ear training for a jazz singer may similarly extend beyond that of a classical singer. Rather than turning it into a moralistic and right versus wrong thing, think of it as different, instead.
Learning to sing classical music may be beneficial or it may be detrimental to a non-classical singer, depending on the instructor and the instructor’s view, dogmatism and pedantry regarding the subject, or a more open minded approach.