I just blogged about this at more length here http://www.aishdas.org/asp/what-did-the-elders-see . The most relevant bits:
Rashi says that they saw something like the Ma’aseh HaMerkavah, the chariot that Yechezkel saw. “And above the firmament which was over [the chayos’] heads looked like sapir stone, the image of a throne; and on the image of a throne was an image that looked like a person upon it above it.” (Yechezkel 1:26) And, in fact, Targum Onkelos on our verse inserts the word “yeqar” to say that they saw the “glory of the G-d of Israel”. This parallels Yechezkel’s description of seeing something that “looked like the image of Kevod Hashem, the glory of Hashem”. (Ibid v. 28)
According to Rav Sa’adia Gaon (Emunos VeDei’os 2:10), there is a kavod nivra – kavod as a created thing. The vision at Mount Sinai and that of Yechezkel were not of Hashem, as that is logically impossible. Rather, they saw this kavod. The Rambam’s approach (Moreh Nevuchim 1:64) is similar to Rav Saadia’s, except that he writes that the phrase “Kevod Hashem” is a synonym; it could refer to either Hashem Himself, in all His glory, or it could be used to refer to the kavod nivra. In our case, the text means that they saw the kavod nivra. However, in Moshe’s later request, he was asking to see Hashem Himself, which is why he was unable to have his desire granted.
Rav Sa’adia Gaon writes that the shechinah is indeed part of the physical world, but that it is a kavod nivra. In fact, Rav Sa’adia Gaon holds that the term “shechinah” refers to any miraculous thing that reminds the viewer that Hashem is shochein beqirbo, dwelling with him. Thus, the pillars of fire and of cloud were the shechinah, as were the vision of Mount Sinai and of the Merkavah. Rav Sa’adia Gaon’s notion of kavod nivra can be a physical object. Therefore this vision could occur through regular, physical sight.
This is where the Rambam’s opinion diverges. He holds (Ibid. 2:6) that the kavod nivrah could only be seen prophetically. It is different in kind to the pillars of fire and of smoke, which were physical entities created miraculously.
The Ramban disagrees with both. In his commentary on the verse where Hashem promises Yaaqov that He will descend with him to Egypt (Bereishis 46:1), the Ramban says that “Sh-echinah” is a name of Hashem, not a created thing (nor a class of them). However, this does not mean that Mosheh and the elders actually saw Hashem in human form. The Ramban on our verse explains that the vision was prophetic. It would seem that in the Ramban’s view, a prophecy can be a vision of something that cannot truly exist.
This indicates that underlying our debate there must be a basic difference in how the Rambam and the Ramban understand prophecy. Even though the Rambam agrees that the vision was prophetic, he still argues that it could not have been of Hashem, because He has no body.
We find an instance of a similar debate in their understandings of the beginning of Parashas Vayeira. According to the Rambam, any narrative that involves people seeing mal’akhim must be the retelling of a prophecy. Mal'akhim do not have physical substance; they cannot be physically seen. Therefore, the Rambam holds that the parashah opens by telling us that Hashem visited Avraham, and then elaborates by telling us the substance of the visit, the prophecy that Avraham received. In other words, Avraham did not interrupt Hashem’s visit to welcome what he thought were three people. Rather, the visit itself was the vision in which Avraham hosted the three mal’akhim. (Moreh 2:42)
The Ramban takes issue with this understanding. After all, did these mal’akhim not then proceed to Sodom where they saved Lot? Was Lot not really saved? According to the Ramban, the story physically occurred. Avraham saw the mal'akhim in the regular sense, actually fed them food, etc… (Bereishis 18:2)
What does the Rambam do with the Ramban’s question? The Abarbanel, in his commentary on the Moreh Nevuchim, writes that according to the Rambam, things seen in prophecy really occur. They are visions of events happening in higher planes of reality. The prophet’s mind and pen may make sense of the vision by interpreting its contents as things familiar from normal sensory experience, but the event seen is both non-physical and real. This is consistent with the Rambam’s position on our verse in Mishpatim. They saw something real. And since G-d does not have a body in any plane of existence, mot even a metaphysical “body”, their vision had to be of kevod Hashem, something created to be a metaphor for them to see.
The Ramban, on the other hand, understands prophecy to be the relaying of a message by the medium of a metaphor. The message relays a truth, but the vision is not of something real, it is a kind of communication. He, therefore, is not bothered by the idea that the metaphor they were given the message in was an anthropomorphic one, that of Hashem sitting on a throne.
In neither case do they actually see G-d -- they either see a created entity that is His Kavod, or they watch a metaphor that relays a message, and in the message, G-d is represented by a man in a throne.
(The parts of the blog post I omitted relate this back to how we see Judaism in general, and the dispute between the Lithuanian schools [yeshiva and mussar] and Chassidus.)