What makes the dance sequence from Pulp Fiction so iconic?

I'll attempt an answer to this question, but I warn you in advance that you're not going to get any single reason. The question isn't really answerable in that way and as such is only open to subjective reasoning.

The first thing to note is that the scene was always meant to be in the movie, regardless of John Travolta's casting (as this article shows):

Many have assumed that the dance sequence at Jack Rabbit Slim's was intended as a reference to Travolta's star-making performance as Tony Manero in the epochal Saturday Night Fever (1977); Tarantino, however, credits a scene in the Jean-Luc Godard film Bande à part (1964) with the inspiration. According to the filmmaker,

Everybody thinks that I wrote this scene just to have John Travolta dancing. But the scene existed before John Travolta was cast. But once he was cast, it was like, "Great. We get to see John dance. All the better."... My favorite musical sequences have always been in Godard, because they just come out of nowhere. It's so infectious, so friendly. And the fact that it's not a musical, but he's stopping the movie to have a musical sequence, makes it all the more sweet.

Tarantino's comments show what he was aiming to achieve. A fun, friendly and infectious dance scene that comes out of nowhere. That's exactly what he achieved.

One minute we're watching these tough gangsters mixed up with drugs like cocaine and heroin and then all of a sudden - dancing! Out of nowhere. On top of that, their type of dancing is very peculiar. Their bodies appear to get more and more into the routine, whilst their faces (and Uma Thurman's in particular) remain so expressionless.

That alone makes the scene both enticing and appealing, as it immediately stands out as being "out of place" in the film (and therefore memorable).

However, the fact Travolta is in the scene, especially given his performances in Saturday Night Fever, make it even more special. Quoting again from the above article:

Jerome Charyn argues that, beyond "all the better", Travolta's presence is essential to the power of the scene, and of the film:

Travolta's entire career becomes "backstory", the myth of a movie star who has fallen out of favor, but still resides in our memory as the king of disco. We keep waiting for him to shed his paunch, put on a white polyester suit, and enter the 2001 Odyssey club in Bay Ridge, Brooklyn, where he will dance for us and never, never stop. Daniel Day-Lewis couldn't have woken such a powerful longing in us. He isn't part of America's own mad cosmology.... Tony Manero [is] an angel sitting on Vince's shoulder.... [Vince and Mia's] actual dance may be closer to the choreography of Anna Karina's shuffle with her two bumbling gangster boyfriends in Bande à part, but even that reference is lost to us, and we're with Tony again....

Similarly from the above article:

Estella Tincknell notes that while the "diner setting seems to be a simulacrum of a 'fifties' restaurant...the twist contest is a musical sequence which evokes 'the sixties,' while Travolta's dance performance inevitably references 'the seventies' and his appearance in Saturday Night Fever.... The 'past' thus becomes a more general 'pastness' in which the stylistic signifiers of various decades are loaded in to a single moment." She also argues that in this passage the film "briefly shifts from its habitually ironic discourse to one that references the conventions of the classic film musical and in doing so makes it possible for the film to inhabit an affective space that goes beyond stylistic allusion."

Time Magazine listed the dance as one of their top 10 movie dance scenes ever, saying:

We already knew John Travolta could dance. He discoed and hand-jived his way to stardom in Saturday Night Fever and Grease. But in the 1980s Travolta’s career fizzled and we sort of forgot about him because — let’s be honest, here — we have pretty short attention spans. But in 1994, Travolta made his comeback in Quentin Tarantino’s stylized crime film Pulp Fiction. His smooth, understated twists to Chuck Berry’s “You Never Can Tell” were so cool, we hardly noticed his weight gain and thinning hair. If we were coked up Uma Thurman, we’d let our Mob-boss boyfriend set us up with John Travolta any day.

So you can see a recurring pattern in all the comments. It appears to be the fact that the scene is so unusual and different from the rest of the movie, making it instantly memorable, coupled with the fact it was a revival of sorts for John Travolta, reminding the world of his dancing talents and evoking memories of his earlier role in Saturday Night Fever.

As said in other answers, this question is hard to answer because it's open to subjective reasoning. I'm still going to try :)

This scene is about constrast, as are lots of scenes in the movie. The whole story goes back and forth from very violent situations to very common day to day stuff. For example the conversation about fast food in foreign countries.

The dance scene is particularly quirky on various levels. All of a sudden, these two people stop a gangster reunion to hit the dance floor and twist together. They do this as normally as they take hard drugs and kill people. Hence the quirkiness. In addition, the fact that Travolta is in this scene (even though the scene was written before he was casted for the role, as said by Andrew Martin on his answer) adds more quirkiness, as he is famous for impressive dancing skills, and is seen here simply twisting casually, without any particularly impressive moves whatsoever. He's just a ganster being cool having a not-feverish saturday night fun.

As the success of the movie seems to be due to it's quirkiness, and this scene is particularly unexpected, it is a powerful sample of Pulp Fiction.

From a cinema goers point of view (not filmmakers/magazines etc)-

I remember at the time Travolta was a washed up actor and Tarantino totally revived his career. As Pulp Fiction became a worldwide success Travolta got an oscar nomination. Here's a very talented man who has finally got back to where he deserves to be by giving a great performance in a great movie. Subconsciously the dance scene had become more than 2 people dancing to a song. It had become Travoltas celebration dance, that he's back near the top.

Just my penneth worth.

You see, I think this scene plays a much more significant role in the events that play out than previously described. During this scene, you can see their infatuation with each other grow to the level of sexual attraction. You can tell with how close their bodies get to one another, and you can also see it in their faces. You notice again when they waltz back into Mia's house and share their "comfortable silence". This is what prompts Vincent to go "take a piss", and have his self reflection of remaining loyal to Marcellus and not trying to sleep with Mia before he leaves. It is of course at this moment Mia overdoses on Vincent's heroin, which leads to the climax of this part of the story. That's just what I think.