I scanned a few replies, and I'm seeing a lot of mention of spotlight balance and niche protection as if it's somehow clearly distinct from PC balance in general. I'm not really sure that's the case. Anything you excel at is likely to get you the spotlight, so I don't really know what the difference is. Perhaps people are talking about combat balance? If that's the case then outside of 4e, classes aren't designed to all be balanced on the combat tier. They are all designed to be able to meaningfully contribute in it, but some are explicitly supposed to be be better at combat in general, and better at certain elements of it in general.
As a DM, I care very much that the there is an overall balance between the PCs--which means that some will be have a stronger contribution in combat than others, and that they will hopefully also play differently.
In fact, for the current long running campaign I recently eviewed some stats I had projected on DPR between the 5 members of the party to see how it looks now that they are 5th level. The reason for that is to see how they were stacking up in that one particular category of contribution, so I could see if it looks like it's in the ballpark it should be in. I was glad to see the numbers came out looking really good. The (not terribly optimized) Battle Master Fighter wins for DPR, followed by the Swashbuckler Rogue. Next up is the Pact of the Blade Warlock with Polearm Master. My home brew Warrior-Mage class comes in fourth, with the Lore Bard who isn't built for combat coming in last. Based on the other ways the characters have of contributing, this seems about right. The lower the DPR ranking, the more special features (like spellcasting) the characters have. The Bard actually has a Fochlucan Bandore, Wand of Magic Missiles and Circlet of Blasting (none of which were included in the DPR calculations), so he has a lot of extra spell power to bring in addition to his general 5e bard coolness.
As a player, I would also care about it, because like most people when I create a character I'm choosing to excel in whatever it is that character is supposed to be good at. If someone else is better than me in "my thing" then I'm not happy. At the same time, if I'm better than someone else in "their thing" I'd be uncomfortable. It's not about competition, it's about each player on the team being the best in the party at doing whatever it is they do. And unless what you do isn't relevant to D&D adventuring, that's generally going to mean (at least in 5e) that the party ends up with a pretty good overall balance between all the PCs.