What is the difference between The sword in the Stone and Excalibur?

There may be no difference, depending on your source. The sWord in the stone story is first found in the Story of Merlinattributed to Robert de Boron. Robert gives no name to the sWord in the stone. He may have intended it to be identified with Arthur's sWord Caliburn or Escalibor (named "Excalibur" by Sir Thomas Malory in his Le Morte d'Arthur) but does not explicitly say so.

The identification becomes explicit in an addition to the Story of Merlin which is today commonly known as the Vulgate Merlin which adds an account of the first five years of Arthur's reign, up to the entombment of Merlin and the birth of Lancelot. See http://www.archive.org/details/vulgateversionof02sommuoft , page 94, line 26 for the Old French version and see a Middle English translation at http://quod.lib.umich.edu/cgi/t/text/text-idx?c=cme;idno=Merlin;rgn=div1;view=text;cc=cme;node=Merlin%3A7 , page 118. Here Arthur fights with the sWord Escalibor which is clearly identified with the sWord Arthur had drawn from the stone.

There is also a late Welsh account based mostly on the Vulgate Merlin which identifies the sWord in the stone with Arthur's sWord, here called Kaledvwlch as is usual in Welsh texts. See http://www.maryjones.us/ctexts/kaledvwlch.html .

However there is an alternate version of Arthur's early years, today commonly known as the Post-Vulgate Merlin. In this version Arthur, with Merlin's aid, obtains his sWord Escalibor from a hand and arm which rise from a lake. See http://books.google.com/books?id=qjk27gRgV1IC&pg=PP7&dq=intitle:Merlin+inauthor:Paris&lr=&as_drrb_is=q&as_minm_is=0&as_miny_is=&as_maxm_is=0&as_maxy_is=&as_brr=0&cd=2#v=onepage&q=&f=false , beginning at page 196. An English translation of a Spanish adaptation of this work is found at http://members.terracom.net/~dorothea/baladro/index.html . Note that in this account the actual finding of the sWord in the lake is omitted with other material that should appear between chapter 21 and chapter 22. But the story of the naming of the sWord appears at the end of chapter 22.

No other early romance tells of the origin of the sWord, save for a statement in Geoffrey of Monmouth's Historia Regum Britanniae that Arthur's sWord Caliburn was forged in Avalon.

Sir Thomas Malory's Le Morte d'Arthur actually derives in its earliest sections from an account which mixes material from the Vulgate Merlin and the Post-Vulgate Merlin. Accordingly in Book I, chapter IX, Malory uses the Vulgate Merlin version in which Arthur's sWord Excalibur is identified with the sWord from the stone. But in Book I, chapter XXV, Arthur obtains the sWord from the lake and in Book II, chapter III, this sWord is said to be Excalibur. See http://www.sacred-texts.com/neu/mart/ .