The same CAPCOM (presumably Charles Duke) is seen for both landing and EVA. In real life, there were three different CAPCOMs during this period. During landing the CAPCOM was Charles Duke, post-landing was Owen Garriott, and the EVA CAPCOM was Bruce McCandless.
Just prior to the lunar landing, the command module Columbia is shown emerging from behind the moon far ahead of the lunar module Eagle. At this point, Eagle had dropped into a lower orbit, which increased its speed and placed it well over a hundred miles in front of Columbia. The show is partly correct, in that Mike Collins, in Columbia, was able to communicate with Earth before Armstrong and Aldrin, due to his higher orbit bringing him above the horizon first.
After initial LEM and CM separation, the ladder on the LEM is shown on the back side of the lander.
The lunar module has a corrugated metal panel on the left front side of the descent stage. This is actually the lunar rover and was not used until Apollo 15 two years later.
When the Lunar Module (LM) "Eagle" is performing it's Powered Descent Initiative (PDI) burn, Lunar Surface Sensing Probes can be seen extending from all four landing leg pads. After Apollo 10, the probe was removed from the forward leg pad, as is was deemed to be a potential spacesuit snag or puncture hazard for the astronauts descending the LM ladder on that leg.
Just before Neil Armstrong places his first footstep on the moon, you can see the outline of his footprint already on the moon surface in exactly the same spot, but without the boot tread details, perhaps from a previous take.
Considering the quality of the special effects and the CGI shots in the rest of the miniseries, the crash of the LLRV seems like something from a cheap 1970's TV show.
During the discussion about not hesitating to abort the issue, there is a package of Lucky Strike cigarettes on a desk. It has a bar code on it. Barcodes were not on products for purchasing until about 1973 and this scene is set in 1969.
As Aldrin is calling out the data during the landing he comes off as being emotional and even slightly frantic. As can be heard on the audio recording of the actual landing, he was calm and relaxed throughout.