The rule on bonus-action spells states:
A spell cast with a bonus action is especially swift. You must use a bonus action on your turn to cast the spell, provided that you haven't already taken a bonus action this turn. You can't cast another spell during the same turn, except for a cantrip with a casting time of 1 action.
I'm wondering what the effect of removing this rule would be.
Sorcerers would become overpowered
Most spells that take a bonus action to cast are relatively minor, like misty step or healing word. They are often utility spells, or spells that enhance a physical attack (like a paladin's smite spells). Most of them wouldn't unbalance play if they were permitted during the same round as another leveled spell.
But whether or not unbalancing bonus action spells exist by default, a sorcerer's Quickened Spell metamagic is highly dependent on the current rules on bonus action spells. Sorcerers can cast Action spells using bonus actions for the cost of 2 sorcery points, which is far more powerful if they could then cast another leveled spell again with their action. This is particularly problematic if you consider the Sorcerer/Warlock multiclass, which can use Warlock spell slots to regain Sorcery points as a short rest resource.
The ability to cast two fireball spells in a single turn multiple times per combat at low levels is game-changingly problematic.