In a "dice pool," one considers each die individually, rather than as a collection of components to be summed. As an example, an attack in the HERO System might do 8d6 damage; to resolve it, you'd roll eight six-sided dice and add them up to produce a number between 8 and 48. In a die pool system, you might see how many of the dice showed a 5 or a 6 on the top face, and count those as "successes" toward a benchmark.
Rolling vs. a target number can be used in both die pool systems and otherwise. In HERO, for example, you roll 3d6 and sum them, trying to stay below 11 (or an otherwise modified target). In Deadlands, you use your die pool and select the highest rolled die, comparing it with a target number modified by the difficulty of the task at hand.
Rerolls are almost always identical to just adding dice.
The only time that rerolls would not be taken as just being additional dice is, to my knowledge, in only two cases: when enough dice succeed to prevent all the rerolls from being used, or when there are too few original dice to make the rerolls practical. If you can reroll failures again, this becomes less significant, but even then you're still doing not much more than doing the addition process again.
I even wrote a script to test this, if you feel like giving it a whirl. In some rare cases it might be possible to game the mechanics to do something other than just adding dice (I'm an English Education major, so take that with a grain of salt), but at that point you'd be designing a mechanic around creating interesting probabilities rather than making a game that rewards rerolls.
You could experiment with using modifier on rerolls or original results, which would produce results distinct from just adding a die, or using different dice, but I don't think that's what you intended.
Rerolls become less meaningful when the target number is low and when the reroll pool is large relative to the total of dice, and are more meaningful when you have a high target number and/or lots of dice. Each subsequent reroll opportunity is exponentially less valuable than the one before it.
Best Answer
My suggestion is to actually discuss this with the players. If you all agree on some range of dice, you can tailor runs according to their abilities.
Shadowrun is a system that is incredibly easy to break. Discussing how powerful you and the players want the characters to be is almost mandatory. Otherwise you might end up with one who might do very little to optimize their characters, while others look for ways to get that one more die to their rolls, creating a massive powergap between players.
Seriously, after my first character, I could easily make melee adepts with 21-22 dice on attack rolls.
Find your preferred "power level". No one wants to be the underpowered sidekick.