Should my Aggressive Assault use a Plasma Rifle or an Alloy Cannon?

I use an aggressive Assault soldier (full right: Aggression, Close and Personal, Rapid Fire, Bring 'Em On, and Killer Instinct) equipped with an Alloy Cannon. She hits hard.

Having now read a lot of questions and answers here, I've seen some people advocating the use of a Plasma Rifle with an Assault class. Perhaps this loadout is really more for a Tactical Sense Assault with a full left skill tree.

Could someone please describe and/or compare the use cases for an Alloy Cannon Assault vs. a Plasma Rifle Assault?

The weapon choice for an Assault really comes down to a decision regarding how you are going to use that Assault trooper. If you are going to walk up beside the aliens and shoot them, choose the Shotgun class weapons. If you are going to spend more time at range, choose the Rifle class weapons. While certain abilities work better with a certain class of weapons, all the abilities work with either. This means that the various Assault builds are not constrained by a specific weapon in order to be effective. In fact, you can change the weapon you are using as the game advances, since once you have Alloy Cannons and Ghost Armor, getting up close and personal with an alien is a viable tactic, and you will do a ton of damage.

As an example of when you want to use an Assault to fight at range, at higher difficulty levels (Classic and Impossible), you do NOT want to fight more than a single group of aliens at a time. The only way to do that is to be very, very deliberate regarding when you reveal more map tiles.

While a close combat specialist Assault does a ton of damage at close range, it is very difficult to advance to close range without revealing more map tiles and therefore, more potential alien threats.

When attempting to fight a single group of aliens at a time, a Plasma Rifle equipped Assault, with Lightning Reflexes, is your primary scout. She makes a single move into unexplored territory. If no aliens appear, she backs up a bit into cover on her second move, being very careful not to reveal any more map tiles.

If aliens appear, Snipers with Squad Sight take their shots. Everyone else goes into Overwatch and the Assault retreats back to her starting cover spot (and may even use Run and Gun to get back and go into Overwatch). The aliens may advance, at which point you deal with them without revealing extra map tiles (which may entail further retreating). If the aliens don't advance, you move the Assault up one move, fire with Snipers and then retreat. Continue until the aliens are defeated.

Played this way, the Assault is always far enough away from the aliens that the range penalty on the Shotgun class weapons makes them less effective than the Rifle class weapons.

In summary, the close combat specialist Assault is great when you either know there are no more enemies (such as when dealing with Ethereals and Elite Mutons on the bridge of UFOs) or you don't care if there are more enemies (such as on lower difficulty levels, or late in the game when you have Ghost Armor). The Rifle equipped Assault is great when methodically clearing a map of alien threats, and with Run and Gun and Rapid Fire, is still effective in the case where you know there are no other threats to deal with.

Finally, to answer the question in your title, your aggressive Assault should be using an Alloy Cannon. However, you may want to consider not using an aggressive Assault.