Neil Armstrong used something for extra safety that nobody else did upon stepping off the lunar module for the first time, because it was the first and they weren't sure what was going to happen.

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First, take this image of the lunar lander: note that the ladder does not reach all the way down. The distance from the last rung to the ground was roughly three times that between the previous rungs, and each astronaut had to "jump" to get down the end. This ended up being part of the procedure for landing, and you can see that in some of the "one small step" videos which start with Neil first jumping down to the LEM legs (not on the surface yet), then jumping back up as a procedural test, to make sure it was possible to return to the ladder safely.

Armstrong: Okay. I just checked getting back up to that first step, Buzz. It's...The strut isn't collapsed too far, but it's adequate to get back up.

The reason the ladder wasn't made farther is that the engineers were playing it extremely safe; they really weren't sure what was going to happen upon landing on the moon, and it was theorized the lander would sink much farther in than it actually did -- specifically they were worried about the "fluffy" powder on the surface called regolith where the nature was unknown. (There was genuine worry from some the spacecraft would be "swallowed up".) To be clear, the depth the legs sunk matched the prediction -- 2-8 centimeters -- but engineers built for extra safety.

It should also be added there was also extra safety built around the softness of the landings, but the pilots did well, and out of all the lunar landings, Apollo 11's was the softest landing at around 0.5 m/s, roughly half that of Apollo 12's.

Due to these unknowns, for Neil's stepping down he had a safety tether. You can see it fairly easily in this training picture.

The Lunar Equipment Conveyer (LEC) is a device which the astronauts will use during the EVA to transfer equipment to or from the ascent stage. It may also be used by the crewmen as a safety tether when moving down the ladder or as an aid in ascending to the ascent stage.

The LEC is a thin 60 foot continuous loop of one inch wide strap, which loops through a support point in the ascent stage and back to the crewman on the surface.

So, even if Neil could not jump back up the ladder on his own, he could get some help with the LEC tethering him to the spacecraft. He was still tethered stepping off on the lunar surface, because again, the effect of the lunar surface was fairly unknown. Any extra "pause" you are sensing likely stems from both the need to manage the extra equipment combined with the general safety worries.

Once Neil safely made it down, there is no reference in the mission transcript to Buzz using the tether, so we can assume that they decided there was no issue without it. No subsequent missions used it, either, although in the Apollo 12 mission right after, Pete Conrad said, upon disembarking from the last rung of the ladder:

Whoopie! Man, that may have been a small one for Neil, but that's a long one for me.

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The One Small Step transcript and safety tether discussion hosted at NASA are both useful for further reading.