What purpose do the luchos serve? By this I mean to say the physical tablets themselves. Had Moshe gone up to Har Sinai and come down with nothing, or perhaps a fully written out torah scroll, would we have been missing anything?
The Alter Rebbe in Ch. 53 of Tanya says the following (parenthesis) are my own additions to help better understand this very complicated idea.
Ch. 53 also mentions how the lack of luchos during the 1st and 2nd Beis HaMikdash affected how the Shechina came down. See Yoma 21 that discusses how the Shechinah didnt dwell in the 2nd Temple because of the lack of luchos. The Alter Rebbe explains in Ch. 53 that the Shechinah during the second temple came down in a gradual descent of malchus d'atzilus into malchus d'beriah, etc.
During the time the First Temple stood, the Ark and the (Luchos) Tablets were in the Holy of Holies. The Shechinah — which is Malchut of Atzilut, (the revealed light of the Ein Sof) dwelled there and was clothed in the Ten Commandments. (This G-dly revelation) was far more intense, and with a greater and mightier revelation, than the same (category of) revelation in the shrines of the Holy of Holies above in the upper Worlds. (Each world has a reflection of the other worlds so there is a Beis Hamikdash in our world may it soon be rebuilt, in the spiritual world of Asiayh, Yetzirah, Beriah, and Atzilus as well abeit in a more spiritual form that has a greater connection to G-dly light ie: the Ohr Ein Sof) For the Ten Commandments are the all-embracing principles of the whole Torah, which derives from the level of Supernal Chochmah, that is far higher than the “world of manifestation.”(Asiyah) In order to engrave them on material tablets of stone, it did not descend degree by degree, parallel to the order of descent of the Worlds, this Material World.
The natural order of G-dly light transformed into this physical world follows a very organized and vast structure known as Seder Histalshelus http://en.wikipedia.org/wiki/Seder_hishtalshelus In the case of the G-dly Light entering into the Luchos, in the Ark, in the Holy of Holies, in the Beis HaMikdash, this structure was skipped and the G-dly light entered unfiltered and to a degree higher than even the highest of high such as the Sefira of Keser. This was why there were unbelievable miracles witnessed daily in the 1st Temple that defied nature and all logic see Pirke Avos Ch. 5:5
Prior to the giving of the Torah at Har Sinai, spirituality could not stay in the physical world. When spirituality was brought down into this world, it was attached to a physical object, like Yaakov and the sticks. Once the object was done being used the spiritual element left back to the supernal world and the sticks turned back to sticks. After Har Sinai the Spiritual and Physical blended together. The Alter Rebbe is most likely expounding on something from the Zohar that describes the highest levels of divine hashpa (influence) being brought down into the world unfiltered and revealed more through the 10 commandements in the ark in the holy of holies than the highest levels of G-dly revelation such as Kesser of Atzilus.
This only addresses what purpose the luchos severed since it seems a Torah Scroll or just an Oral Law would be just as good.
For a more historiological answer than @mochinRechavim's, see this article by Dr. Meshulam Margaliot at Bar-Ilan.
Dr. Margaliot pursues this question along with R Nehemia's position in Shemot Rabbah 47.6 which agrees with the Sages' position in Mekhilta of Rabbi Ishmael (Tractate de-ba-Hodesh 5):
How were the Ten Commandments given? Five on one tablet and five on the other. 'I am the L-rd' written across from 'You shall not murder'. This is according to R. Hanina b. Gamaliel, but the Sages say ten on one tablet and ten on the other.
He answers that the Ten Commandments were given in the form of a traditional ancient Near Eastern contract. Each party gets their own copy of the agreement. He says that of course G-d's copy lives in the aron since that is where the Shekhina lives...
But what about the Israelite's copy, on the second Tablet, where was that copy placed? Here we note a common practice in the ancient Near East. When a treaty was made between parties of unequal status, the lesser partner, or vassal, would place his copy of the pact in the temple of his god, the reason being that the vassal had then to take an oath in the name of his god to "the great king." (See Ez. 17:11-19. The reference here is to the king of the Hittites, who made treaties with the rulers of smaller kingdoms in northern Syria during the first half of the first millennium B.C.E. This custom, however, undoubtedly dates much further back.)
Since the Israelites had the status of vassal vis-à-vis G-d and were the lesser partners to the Covenant, it was reasonable for them to file their copy of the Pact in the Holy Ark of the L-rd their G-d. Thus we conclude that both Tablets were placed together in the Ark in the Tabernacle, and later in Solomon's Temple: "There was nothing inside the Ark but the two tablets of stone which Moses placed there at Horeb, when the L-rd made [a covenant] with the Israelites after their departure from the land of Egypt" (I Kings 8:9).
The reason for the luchot, then, is that they define the covenant between the King and the vassal kingdom, i.e., Israel. They are the formalization of that relationship and it reflected the practices of the people in the area of the time regarding covenants.