Dice systems are, at their heart, kind of dull. This is probably a good thing, since most of the difference between systems can be built upon their framework, so you want them to be simple, reliable and dull. This is on my mind for reasons that are probably a different post, but I've been chewing on a core dice mechanic.
Right off the bat you have three big camps of dice mechanic. You have the flat roll, the tally, and the count. Flat roll is most famous in the various incarnations of D&D, which has used it as both roll vs. moving target (old D&D THAC0 tables) or roll + bonus vs target (4e). The "flat" roll may actually have a curve (as in the case of Dragon Age's 3d6) but it's always the same dice, and differentiations in skill are represented by changing the target number or changing the bonus. Fudge dice are another weird example of a flat roll that don't necessarily look it because the range is wacky (-4 to 4) but it's a fixed set of dice all the same.
Tally systems, such as those used in WEG's Star Wars or AEG's Roll & Keep system, are based on totaling up a variable number of dice. Variations in skill can change the size of the die pool, and while there may be some extra mechanical fiddliness in terms of how many dice are counted (as in the Case with Cortex+ or any game with bonus dice) , the core idea is that the pool of dice is the variable and as a subsidiary idea, the size of the dice may also be a variable.
Count systems are usually considered success counting systems, like Storytelling or Burning Wheel. You roll some pool of dice (of variable size, like a tally), but rather than add up the dice, you count the number of results that hit some particular criteria (such as 7+ on a d10, or 4+ on a d6). Left purely in that form, this is just a highly specialized tally system (effectively the dice have some number of 0's and some number of 1's) but it's worth differentiation from the tally because the ways it establishes differentiation can include changing the rules of success counting. For example, successes might usually be on a 7+, but in this particular area in which you excel, they might happen on a 6+. Thus, while the size of the die pool may be one axis, another will often include the means of determining successes.
There's a lot more fiddly in this. You can add bonuses to a Tally system to make it feel a little more flat-like. In any system, you can add variations in how you measure success and how you handle things like critical successes and failures (all to say nothing of rich rolling). You can get into wacky hybrids or edge cases (like set building), but those three models really cover the bulk of approaches, and they have different strengths and weaknesses.
Flat rolls are the simplest. Even when they require some math, it is usually quite simple, and perhaps more importantly it's _perceived_ as simple. Reading the die results in a flat roll is easy, with almost no learning curve. Even if the "post-processing" of adding bonuses or the like takes soem effort, it is -after- the roll, a critical point of distinction.
Tally rolls are probably the most robust. If you want to hang a million different mechanics off a roll, or make the die rolling a bit of a game in its own right (like rolling lots of dice, then doing many mini-resolutions from the pool) then this is probably the approach to take, but it absolutely runs the risk of daunting players. Even if you don't do a lot of fiddly stuff, the perception is that math is hard and slow, and tallying up the dice creates that sense of friction. This can be mitigated with small or familiar dice pools, but it's always the specter over the system.
Counts are something of a compromise. They offer much of the same mechanical robustness of Tallies, but they promise greater simplicity than doing math, and in my mind they deliver on that, at least to an extent. There can be a little bit more of a curve in picking up a count game, but the act of reading the dice is an educational one, and most people get much faster at it with only a little practice. Unfortunately, there are some problems that come with that. First, that simplicity is based on the method of counting remaining the same, and if the system leans on changes to that, it slows things down. Also, you can only rely on the simplicity scaling so far - if the dice pools get huge (like, Exalted huge) then it will still bog down.
All three methods can work very well. Even more, all three are robust enough that if there's some particular element you want to accentuate or avoid then you can easily tweak them to that end.
I'm dwelling on this because I have a dice system in mind for a project, and i want to step back and consider its weaknesses and strengths before I totally buy into it. More on that tomorrow.