First note that wüst usually means something like "wild" or "chaotic": eine wüste Schlägerei -- a wild brawl, eine wüste Bedrohung -- a savage threat, etc. But wüst does have a meaning which overlaps that of öde, namely that of "desolate" or "deserted". (The DWDS entry for wüst simply defines it as öde in this sense.) My impression, based on the usage examples I've seen, is that the difference is mainly in size and reason. The word öde connotes abandoned; no one is there because they all left; a house, a street, a neighborhood might be öde. For example (via the DWDS usage database):
Das öde Haus tönt von unsrer glücklichen Jugend. (Voß, Richard: Zwei Menschen, Stuttgart: Engelhorn 1911 , S. 50) -- "The barren house resounds with our happy youth."
Aber um 11 Uhr sind die Straßen völlig öde und leer. (Die Zeit, 21.04.1949, Nr. 16) -- "But at 11 o'clock the streets are completely deserted and empty."
Meanwhile wüst connotes wasteland; the entire region is incapable of supporting life:
Wie sich überhaupt alles, was zumindest etwas freundlicher klingt in diesem wüsten, ja nihilistischen Stück, bestenfalls als Erinnerung an vergangenes Glück hören lässt. (Die Zeit, 11.03.2009, Nr. 12) -- "How everything that sounds at least a little friendlier in this desolate, nihilistic piece can at best be heard as a reminder of past happiness."
But there is considerable overlap in that wüst can mean wilderness in a certain Biblical sense:
Zum Schluß jedoch ist wieder jene Lieblichkeit ins wüste Land gezogen, von der am besten holde Sängerknaben singen. (Die Welt, 10.08.2005) -- "In the end, however, that loveliness has again moved into the wilderness, of which fair choirboys sing best."
One can also speak of a wüste Insel -- "desert island" which presumably means an island where no one is living, not an actual desert. Also öde can mean barren:
Aber ohne Ornament wäre die Welt so öde wie ein Plattenbau. (Die Welt, 17.05.1999) -- "But without ornament the world would be as barren as a prefabricated building."
And finally the two words are often used together:
Die Erde aber war wüst und öde, und Finsternis lag auf der Urflut. (Die Zeit, 04.01.2005, Nr. 02) -- "But the earth was chaotic and barren, and darkness lay on the primeval flood." (This seems to be paraphrasing Genesis 1:2: Und die Erde war wüst und leer, ...)
The DWDS usage database is a good tool for researching this kind of thing since you can compare side by side which words are more strongly correlated with each choice. Note, however, that in this case the database interprets the last name Wüst as an adverb, a bit like that Abbott and Costello routine, so it pays to be skeptical of the results.