The 501st employ tactics beyond "shoot enemy until dead" like other regiments we see during Order 66, mainly because they are uniquely experienced in this fact. They had to deal with rogue Jedi Pong Krell during the Battle of Umbara, with Rex and Jesse both there. Of course Ahsoka is going to struggle getting away from him, Jesse helped capture Krell using different tactics.
In Rebels, Rex introduces himself as Captain Rex despite the fact in the last days of the Clone Wars, he was Commander Rex. In this episode, before Jesse attempts to execute him for "treason" against the Grand Army of the Republic, he demotes Rex of his position as commander and Rex jokes he didn't like to be commander to begin with. So technically, he was back to being Captain Rex. That and the tragedy of the 332nd Company would be too painful for him to bring up anyways, which is why he introduces himself as being from the 501st instead of the 332nd.
The 332nd is still a division of the 501st though. As Anakin’s right hand man, Rex would still have been the de-facto commander of the 501st Legion if he wasn’t presumed dead at the end of the episode, and Appo was likely a Lieutenant or Captain during the Jedi Temple Raid, and promoted to Commander after the fact. Alternatively, this could be a situation not unlike that of Commanders Ponds and Neyo of the 91st, with one assigned to Mace Windu, and the other Stass Allie. Anakin had hoped Ashoka would return to the order before Order 66, so he could have intended Rex to become her right hand man.
According to Jesse, Sidious specified Ahsoka as a target for Order 66, likely because he knew Anakin's former Padawan was a loose end that could come back to bite him if he didn't address it. However, this also explains why Maul was targeted by the order, despite having no affiliation with the Jedi Order at all. Sidious specifically marked him as well, as his former apprentice as a loose end he needed to dispose of as well.
The parameters of Order 66 laid out by Rex in the previous episode showcase just how far ahead Sidious planned the Jedi Purge. He had the clones specifically refer to the Jedi targets by their names rather than their current status as Jedi, meaning that the Jedi cannot avoid execution by just leaving or renouncing the Order. And to prevent any clone from seeking non-lethal methods or questioning the order itself (which could happen given the malfunctioning of Tup's chip), Sidious placed disobedient clones on the execution list along with the Jedi. It's not surprising considering he's a politician.
Not to mention that Tup did not target Anakin as a Jedi when Order 66 was activated in him early on, hinting that the order was more about Sidious enacting a purge of those who could oppose him, and not about literally targeting all Jedi as Anakin was clearly never meant to be attacked. This could also explain why no amount of reasoning Rex could use against Jesse would work, as the Clones are Brainwashed and Crazy to attack anyone on Palpatine's hit list, not rationally carry out an order against Jedi who had supposedly gone against the Republic.
Ahsoka and Rex are only able to find a single working Y-Wing to escape on. This is likely because the majority of the 501st's fighter compliment was sent to Coruscant with Anakin and Obi-Wan, leaving only a few craft undergoing maintenance on Ahsoka's ship.
Which, in retrospect, also explains why the Republic force sent in to storm Mandalore was comprised mostly of gunships, which are far more suited to attacking a planet than getting involved in a humongous space battle.
It's rather fitting that the fighter Ahsoka and Rex escape on is a Y-Wing, considering how the Y-Wing would go on to become one of the iconic fighters of the Rebel Alliance. The Y-Wing's presence in this episode effectively bridges the gap between the Old Republic and the Rebels who would go on to become the New Republic.
Ahsoka is shot in both shoulders and the thigh by weapons capable of breaking apart a battle droid... And while painful, she only suffers minor scratches rather than losing her arms and leg. This neatly explains why the Clones use heavy firepower, including artillery fire, in executing the Jedi Purge, and continue shooting the fallen Jedi: they're just that tough.
Fridge Horror
Did Ahsoka know Padmé was pregnant? If she found out later would she assume she died protecting Anakin?
Maul demonstrates what a cut-rate, broken Sith Lord is capable of. The galaxy doesn't know the difference between the Jedi and the Sith because the Sith kill everyone.
