I never thought all my research on this topic would be useful. This is my moment! When first applying to grad school I actually applied to a phd program for plasma propulsion. Didn't get in but I love the field.
They're similar to to conventional electric thrusters in the fact that they use electrical power to accelerate the propellent as opposed to chemical reactions. However Ion drives, Hall thrusters, and Plasma thrusters all differ on what parts of electrodynamics they use to achieve this and technically all 3 of them create plasma in the chamber and it's neutralized at some point along the way.
I wrote a paper on electric propulsion for my physics E&M class in my undergrad so I can upload that too if you're interested but it's basically a bit more detail about the differences.
Basically gridded Ion drives use a large E field to ionize the propellant and because the positively charge gas is repelled from the E field plate towards a charged grid which the ions pass through where then the exhaust is then neutralized with an electron gun. This has negative effects due to lifetime of that grid because of the many impacts by charged particles.
In the Hall thrusters the gas is still ionized by a large E field(this is the similarity) but the plasma is accelerated by B fields due to the right hand rule out the nozzle. In this case I think an electron gun isn't necessary and definitely not the grid, so these thrusters are all the rage right now because they have a higher life time of a traditional gridded ion drive.
The best example of what we would call a "plasma drive" is VASIMR which stands for Variable something something something magnetohydrodynamic something. It's been too long and a simple google would solve that but I thought that sounds funny. At any rate, in these thrusters the plasma is created by microwave radiation. The plasma is then controlled and modulated using "plasma physics" and shoved little by little towards the nozzle with asymmetric frequency modulation. Basically move 1 to left, 2 to the right. Eventually you leave the right side of the engine. Using this technique you can choose how long you keep the plasma in the 'combustion' chamber. This is the entire advantage of this engine type. The longer you keep the propellant spends in the chamber the hotter it gets, and the higher its exhaust velocity. This increases the specific impulse of these engines at the cost of thrust. On the other hand, you can modulate faster and have more propellant leave per unit time. More mass flow means more thrust at the cost of a cooler propellent so lower ISP.
These engines allow you to dynamically change your thrust and specific impulse just by changing the frequency inside the engine. It's cool shit. This means you could have high thrust when you want to leave/enter an area fast(such as the initial thrusting phase when leaving or entering a new sphere of influence) and switch to lower thrust by higher efficiency during the transfer phase.
To finally answer your pros and cons of electric engines:
Pros: High efficiency due to large specific impulse
Cons: Expensive electricity wise. You need kV and a lot of current to run these suckers. In addition to that low thrust trajectories are typically much longer than a traditional trajectory with a chemical engine due to the time it takes to impart significant deltaV. I just wanted to reemphasize how much power these require...especially vasmr. It would need a nuclear reactor or an insanely huge solar panels to run. We have hall thrusters in use and the dawn mission actually used a set of 3 hall thrusters through the course of its missions, but even these 3 tiny thrusters require like 20m long solar panels to accelerate a relatively light satellite. To use these on human bearing spacecraft to get people places in a reasonable time requires a lot more power.
VASIMR is the Variable Specific Impulse Magnetoplasma Rocket. I should have remembered that.
Let me know if you have any more questions!