Do they both mean a brief version? Eg.
They gave us the _ version of what had happened so far.
And is "break down" related to these words? Like in breaking down the steps for an activity.
They are very similar, but have slightly different goals. Both involve a transfer of needed information.
Rundown is a high-level list, helping a person who just entered the situation to be better acclimated to what is going on without being bogged down with details. "He came to the ballgame late, so I gave him the rundown of the score and batting order."
Lowdown implies privately-held and true knowledge that's being held from the general populous for whatever reason. This is often accompanied by a lowering (quieting) of the voice, as if to convey a secret. "Everyone thought that Jimmy painted the mural, but Jane gave me the lowdown: it was actually her."
It seems so according to Oxford Dictionary
An analysis or summary of something by a knowledgeable person.
informal
The true facts or relevant information about something.
A brief statement or account of the main points of something.
[informal] low-down, recap