Yes, let's consider Charisma for a bard - 12 versus 14 versus 16 (I don't know why everyone is so focused on that one cut-off, there is nothing more special about 16 over 14, than 14 over 12 - they're all obtainable from the core game's recommended generation methods with either fixed or floating ASIs.)
- Bardic inspiration will be 1 use/rest, 2 uses, 3 uses. So 14 has 200% of the uses of 12. 16 has 150% of the uses of 14.
- Features that use bardic inspiration, like cutting words or mantle of inspiration, are all impacted.
- Charisma is also the bard's spellcasting ability, so their spells will have better DCs. Say a foe would save 8/20 times, with 12 they save 7/20, with 14, 6/20, with 16, 5/20. 7/8 is a 12.5% change. 5/6 is a 16.7% change, in number of times that a foe saves out of the number of times they would have saved otherwise.
- Same impact on the high-value Persuasion skill, and because the social interaction system uses thresholds, sometimes this will be the difference between not possible, to possible.
- The bard with gain 1, 2, or 3 spells in mind.
I find it visible in gameplay at the table, the difference between Bards with 12, 14, 16 or for that matter 18 or 20 in Charisma. But the ASIs will typically allow the modifiers of two abilities to be bumped upward. So the half-elven bard might also experience a reasonable improvement to their Dexterity, which will mean similarly better initiative, attack, damage, armor class, and perhaps a key skill like Stealth.