Considering a character is using the same weapon in both of the following scenarios, which would result in a higher damage per round?
The character makes a single attack with advantage.
The character makes two attacks: a regular attack and an attack with disadvatange.
TL;DR Choose 1 attack with advantage, if the difference between the AC and the attack modifier is at least 11, choose 2 attacks otherwise.
For those interested in math, let's calculate the expected values. The following calculation assumes that the expected damage is positive.
Let
$$ \begin{align} E_{adv} &=& \left(1-\left(\frac{(D-1)}{20}\right)^2 \right)\cdot E + \frac{39}{20^2} \cdot (E_{crit} - E)\\ &=& \frac{400E-(D^2-2D+1)E + 39 (E_{crit}-E)}{400}\\ &=& \frac{(360 - D^2 + 2D)E + 39 E_{crit}}{400}\\ E_2&=& \frac{21-D}{20}E + (E_{crit} - E)\frac{1}{20} + \left(\frac{21-D}{20}\right)^2E+\frac{1}{20^2}(E_{crit}-E)\\ &=&\frac{420E-20DE+20E_{crit}-20E+(441 - 42D+D^2)E + E_{crit}-E}{400}\\ &=& \frac{(420-20D-20+441-42D+D^2-1)E + (20+1)E_{crit}}{400}\\ &=& \frac{(840-62D+D^2)E + 21E_{crit}}{400}\\ E_{2}-E_{adv} &=& \frac{(480-64D+2D^2)E - 18E_{crit}}{400}\\ &=& \frac{2E}{400} \left(D^2 - 32 D + 240 - 9 \frac{E_{crit}}{E}\right) \end{align} $$ Using the p-q formula we get $$ \begin{align} K & := & \sqrt{16 + 9\frac{E_{crit}}{E}}\\ E_{adv} \geq E_{2}&\Leftrightarrow & 0\geq E_{2}-E_{adv}\\ &\Leftrightarrow & 16 -K \leq D \leq 16+K \end{align} $$
About \$\frac{E_{crit}}{E}\$ we know that it's at least 1 (no dice involved, e.g. unarmed attack) and assuming the damage before adding the dice rolls is not negative, it's at most 2, which means \$\sqrt{15+9} = 5\leq K<6 = \sqrt{16+9\cdot 2 + 2}\$ and therefore
you should choose the one attack with advantage, if the difference between the AC and your modifier is at least 11.
In short, if the difference between your attack roll modifier and the target AC is 9 or greater, use the one advantage attack
This works because the more likely you are to hit, the better using two attacks becomes. This is because the advantage has less of a chance of mattering as the normal and disadvantage attacks have a good enough chance of hitting even without advantage.
However, as you approach very high Armor Classes, this starts to be less impactful. This is because you are mostly relying on critical hits to deal damage and while Advantage is still better (since you have two chances to score a critical hit as opposed to 1 and then a half-chance), the degree to which it is better is lessened.
Here is a simple chart showing what is best at which to hit bonuses and which Armor Classes. "Adv" means the 1 attack with advantage is better, "TWF" means using two attacks is better even though one has disavantage.
\begin{array}{c|l|l|l|l} \text{Armor Class →} & 8-11 & 12 & 13 & 14 & 15 & 16 & etc. \\ \hline \text{To-Hit Bonus ↓} \\ +3 & TWF & Adv & Adv & Adv & Adv & Adv \\ +4 & TWF & TWF & Adv & Adv & Adv & Adv\\ +5 & TWF & TWF & TWF & Adv & Adv & Adv\\ +6 & TWF & TWF & TWF & TWF & Adv & Adv \\ +7 & TWF & TWF & TWF & TWF & TWF & Adv\\ etc\\ \end{array}
The full table and raw data for this analysis can be found in this Google Sheet. You can download a copy to see the formulas. These numbers represent the multiplier applied to the weapon damage dice. However, do note that the critical damage would not be quite as high, which may change the results (without knowing the ability score modifier, it is impossible to know what portion of the damage will be affected by a critical hit).
