In a game I'm playing, I have a Swashbuckler rogue, and my DM gave me the okay on using two-weapon fighting with a rapier and a dagger instead of two shortswords. (But not rapier and shortsword, because that would be directly better.) I also don't want to use the Dual Wielder feat to fight with two rapiers, because of personal taste. What are the advantages and disadvantages of fighting with a rapier (1d8) and dagger (1d4) in comparison to two shortswords (d6+d6)?
If you use both your action and your bonus action to attack, then there is no difference.
Since both TWF attacks have the same to-hit chance (if you're attacking the same enemy), the average roll is the same regardless of whether you roll $$ 2d6 \rightarrow 2 * 3.5 = 7 $$ or $$ 1d8+1d4 \rightarrow 4.5 + 2.5 = 7 $$
However, if you only have your action (or only 1 attack / turn for some other reason), then the rapier/dagger combo has the advantage that you can choose to use the rapier for this single attack, increasing your average damage when you hit by 1.
Additionally, you can choose to throw the dagger, should the need arise. With your free object interaction, you can then draw another one to be "fully equipped" again.
Furthermore, a dagger and a rapier only weight 3lbs total, while two shortswords weight 4lbs. Two shortswords, on the other hand, are 7gp cheaper than the dagger/rapier combo. Admittedly, however, these are very minor benefits that cancel each other out (imho).
The Rapier + Dagger combo is better
Since your DM has effectively made the Rapier have the Light property, it has the same weight and properties as the Shortsword, but it deals more damage. However the restriction of being required to also wield a dagger is there so let's compare a few things.
They have identical damage ouput
In terms of damage, you have an equal number of opportunities to activate Sneak Attack (two), and the Rapier + Dagger deals an average of 1d8 + 1d4 = 4.5 + 2.5 = 7.0 while the Shortsword + Shortsword deals an average of 1d6 + 1d6 = 3.5 + 3.5 = 7.0
We can see that these deals the same amount of damage on average, even accounting for Sneak Attack.
The Dagger can be thrown, unlike the Shortswords
The dagger provides an additional means of attack through it's Thrown property. Wielding two Shortswords, you could only attack at melee range, but the Dagger changes this, making it the superior choice.
Technically, there's a weight and price difference
Additionally the dagger does weigh 1 pound less than the Shortsword so you would be carrying one less pound (though this likely does little-to-nothing; it's a single pound after all). Additionally the Rapier + Dagger combo costs 27gp while the Shortswords cost 20gp, this could be good or bad depending on whether you want to buy or sell the weapons (such as once you get magic versions), but the difference is very slight.
If you aren't using Two-Weapon Fighting the Rapier + Dagger is still better
In this case you can use your Attack action on the higher damage Rapier, using your bonus action on something besides the dagger. This is strictly better than if you had used your Attack action on the Shortsword. Note that this same logic applies if you get Extra Attack or any means to make more attacks on the same turn (such as the haste spell).
Critical hits are possibly different
The damage of critical hits is equal only if you crit using the Rapier as much as you crit using the Dagger. Anything that increases only the rapiers crit chance, makes the Rapier + Dagger combination better than two Shortswords.
One way this might happens is if you have advantage but only on your first attack (perhaps through the invisibility spell, or from Hiding); attacking with the Rapier for this one attack and then the Dagger would be stronger than if you'd attacked with two Shortswords.
Metagaming / damage order
There is one final thing, though it does not make either option better, it is just a difference between them and it won't come up at most tables as it requires greater knowledge of enemy HP than most DMs give out.
Imagine an enemy has 7-8 health left, (ignoring modifiers etc...), you could kill them only if you were using the Rapier, allowing you to damage two separate enemies, whereas with two Shortswords, you could only ever damage one.
Now imagine an enemy has 5-6 health left, (ignoring modifies etc...), you could not kill them if you were using the Dagger, meaning you would kill the enemy with the Rapier and deal 1d4 a second enemy, whereas with the two Shortswords, you would deal 1d6 to the second enemy.
In the long run, the weapons you have won't play a huge part in your overall dps. There are several other factors that have larger, long term impacts to damage
As a rogue, you can sneak attack once each turn. Using TWF gives you two chances to get your sneak attack in. At level one, it's 1d6. At level 3, it's 2d6. At this point, your sneak attack is averaging the same as a hit with both of your weapons. It only gets higher from here.
Character Ability Scores and Proficiency
As you level, unless you've already maxed out your dex, your ability scores can and should increase, as well as your proficiency bonus. These greatly increase your odds of hitting your target, which also increases your odds of getting your sneak attack. It also increases your raw damage per hit.
Part of what's really good about the Swashbuckler is that you get more options to gain sneak attack on a target, which GREATLY increases your dps. Positioning is absolutely important for a swashbuckler in order to get your sneak attack more often.
So, my recommendation is to play with whatever FEELS correct for your character. The damage on the die won't matter much over time.