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.

Spellcasters with misty step would become even more slippery

If a spellcaster ends up in melee range, misty step is a cheap, guaranteed option to get out without getting slapped by an attack of opportunity. It's basically a Dash and Disengage in one bonus action (and more, depending on what the teleportation lets you bypass). The only downsides are using a bonus action (if you even have any use for it), the spell slot (2nd level, pretty cheap), and the bonus-action spellcasting restriction.

With the restriction, a spellcaster has many options for what to do about an attacker in melee range, but they all have consequences. Disable/push them with a spell? Might come at a cost of using another spell on another target, and could fail if they make their save. Using misty step is a great option, but again may come at the cost of using an important spell. With the way it is, even a melee attacker that hasn't gotten a chance to attack the spellcaster yet (they dashed up or something) can have a significant impact.

Without the restriction, you could try to disable/push them with a leveled spell, fail, and then misty step away once you see it didn't work. Or you could go ahead and cast your more important spell on your more important target and still avoid taking any damage from the melee attacker.

It basically adds this layer of safety that changes the counterplay between spell casters and melee attackers, and reduces the impact of a melee attacker getting into range of a spellcaster.

All-out elimination of the limit would be OP, but you don't have to go that far

However, one could readily fashion a more moderate table-rule that avoids these drastic consequences. Consider limiting one or both of the spells cast on the turn to 1st level spells, or non-damaging-spells, or spells with the Ritual tag, and/or spells of just one particular school (e.g. Illusion). You should be able to loosen up the RAW without destroying play balance.

In a somewhat similar case, at my table, it has never destroyed play balance to let players take a feat allowing them to use a bonus action to cast any non-damaging cantrip that only creates a sensory effect. While that's not exactly the same option you are proposing, it does suggest that one can grant a bit more freedom in casting without unbalancing the game.