It's complicated.
The difference between different dice mechanics is largely about the kurtosis, although some such as D20 and percentile dice behave essentially the same in this regard.
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Fig 1. Random chart off the interwebs showing distributions with different kurtosis.
Figure 1 shows some probability density functions with differing kurtosis. Kurtosis drives the likelihood that a roll will produce extreme results in the tails. Dice with high kurtosis are more swingy. The most swingy mechanics are D20 and percentile dice (which behave as if they were a single 100-sided die). These have a flat distribution where every possible outcome has the same probability.
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Fig. 2. A flat distribution such as a D20 or D100.
Different dice mechanics have different probability distributions. 4DF, for example, has relatively low probability of rolling anything outside the range of +2 to -2. There is about 6% chance of rolling higher than +2. This makes the game dependent of the FATE point economy as the dice roll outcomes are largely clustered around the middle in the range +2 to -2. D&D's D20 mechanic, however, is just as likely to roll a 1, 10 or 20, giving it a much higher probability of rolling in the extremes. Traveller's 2D6 mechanic sits somewhere in the middle. Exploding dice mechanics complicate the calculation - Savage Worlds is famous for being fiddly to calculate the odds for dice rolls.
Generally the more dice involved the lower your kurtosis. One issue with buckets-of-dice systems is that the kurtosis changes with the number of dice being rolled.
Having said all that I have a soft spot for 2D6, as it lends itself to quick mental arithmetic and bonuses of (say) +1 to +3 have more-or-less sensible effects across a wide range of target rolls, which is a nice happenstance. It's more swingy than 4DF and less swingy than D20.
I'll also add some observations about dice mechanics on systems for PbP use.
I'm running a Scum and Villainy campaign on a forum at the moment, and the dice mechanic is slightly clumsy as you have to have some back-and-forth between the DM and players to even determine how many dice to roll. For a game to work nicely in PbP I would suggest a dice mechanic with the following attributes.
- Always rolling the same dice - the difference is the target.
- Bonuses can be added after the fact. If you have metacurrency then it can just be added later.
- Avoid contested rolls. Keep it to the player making one dice roll - through a roller app or otherwise. The DM can subtract the opponent's bonus from the roll.
This approach means that the player can roll straightaway and the GM can frig the roll with whatever bonuses apply. On a PbP game that can save a day or two of turnaround time for a single roll.
2D6 + bonuses would work fine for this, as would 4DF or a D&D style D20. Dice pools where you might vary the number of dice being rolled add a round trip to the conversation. This also includes D&D's advantage/disadvantage mechanics if it's not always obvious when it applies.