Selbständig and selbstständig seem to share the same meaning. What are the differences in tone and usage?
"Selbständig" is the traditional orthography of the word (double "st" was collapsed). It also resembles the pronunciation more closely, since saying a double "st" is hard, even for a German.
"Selbstständig" is the more modern variant that is allowed since the orthography reform. It is also the "more logical" variant, since it really is a composite adjective consisting of selbst and ständig.
There's no difference. Both are one and the same word, and there's also no difference in tonation. But "selbstständig" is the preffered way to use. As you can see in the "Duden", "selbständig" is just an alternative form for the word.
selbständig and selbstständig are two words with different pronunciation but no difference in tone or meaning whatsoever.
Long answer (also addressing some questions, that were only asked in a „duplicate“ of this question)
First of all, there are two possible origins of this words: selbst + ständig and selb + ständig. The latter uses the particle selb which is also used in other German words like selber or selbst. It is most probably impossible to tell which of these two is older.
Secondly, despite many people considering this a spelling issue, it is not. For non-loanwords, like selbständig, all German orthographies (from new spelling over failed spelling reforms to very old spelling) are mainly phonetic and only operate within a certain range of tolerance, such that the following holds (taking into account all orthographies at once): A given spelt word may only represent a very limited number of spoken words and vice versa, e.g., Maß may only represent the pronunciations [mas] and [maːs], and [maːs] may only be spelt with maß, mass, masz, mas, mahs, maas, maahß or other combinations of the elements used therein (ignoring capitalisation).
So, unless an orthography chooses to ignore several major spelling paradigms for this single case only (see below), there is no way that the two pronounced words [ˈzɛlpʃtɛndɪç] and [ˈzɛlpstʃtɛndɪç] are represented by the same spelling. Also, the other way round, any non-sloppy pronunciation of selbstständig has to contain at least two more phonemes than for selbständig. Admittedly, [ˈzɛlpstʃtɛndɪç] may be reduced to [ˈzɛlpʃtɛndɪç] without the speaker noticing, but the same also holds for some other words or pairs of words like Schwellenwert/Schwellwert, the difference between which nobody would consider a spelling issue. (Note that one still might consider such differences an issue of style.)
Another argument for are constructions, where the words are decomposed:
selb- und eigenständig
selbst- und eigenständig
Here, there is a clear difference in pronunciation and I do not think that anybody would consider the difference between the two a spelling issue.
However, over some decades prior to 1996, the Duden dictionary was the official orthographical authority and therefore it could declare single-word rules and indeed did (sort of):
selbständig: Das Adjektiv selbständig ist mit dem Stamm des Pronomens selbst, also mit selb- gebildet. Die heute allein gültige Schreibweise des Wortes ist daher die mit einem -st- (Silbentrennung: selb-ständig). (Duden Bd. 9, 1985, quote according to Ickler)
However, one can still argue that, even then, selbstständig was orthographically correct, since a German speaker could create a new word selbstständig as a composite of selbst and ständig impromptu, which is not addressed by the above quote and therefore orthographically correct (just as one can create the orthographically correct word Hundekotbeutelspender impromptu). Also, one could argue that this is not an orthography issue and therefore was not under the “jurisdiction” of the Duden’s orthographical authority (or any other orthography).
With the orthography reform of 1996, the Duden’s questionable single-word rule was not adopted, so there is no orthographical argument against either word anymore. It should be noted, that the current spelling rules use selbständig and the Duden’s position regarding selbständig was criticised also by Theodor Ickler, a prominent opponent of the orthography reforms of 1996 and 2006.
Finally, which variant is preferable? Historically and etymologically, both are justified, and neither variant is more logical or orthographically preferable (see above). Selbständig is easier to pronounce and hence most probably the dominant word in spoken language. One might also consider it to be easier to read. Selbstständig on the other hand is more consistent with other words beginning with selbst, such as selbsttätig, selbstredend or selbstvergessen.
References and further reading:
There's no difference in the usage of those words. In fact, they are different spellings of the same word.
This seems to be a case of simplification of a word, that's combined of two other words (selbst and ständig). The combination is a bit difficult to write and pronounce. So the colloquial just "merges" both st and this has also been transfered to the written language – both are valid spellings.