Lavarche_, who I am assuming is a native speaker of Turkish, provided an answer to this already, and it may have fully satisfied your curiosity, but if not, I will add more here (and if a native speaker wants to dispute/debate any of this, feel free).
The first source I turned to was Wiktionary. It may not have the clout of the Turkish Language Association [Türk Dil Kurumu (TDK )] or even WordReference, although Wiktionary gives the latter a run for its money in terms of the features it offers (and Wiktionary doesn't have ubiquitous ads that soak up tons of bandwidth). Be that as it may, according to Wiktionary, "mevcut" is a Turkish translation of the word "obtainable."
"Müsait," on the other hand, is derived from the Arabic word for "assistant" or مُسَاعِد.
Translated into Turkish, it means "suitable."
According to WordReference, however, "mevcut" can mean "available" as in "Is this shirt available in any other color?" (just as lavarche_ posted), but it can also mean "present" as in attendance -- Is everybody present for the meeting? or "actual" as in "This movie is based on an actual court case."
It has other meanings/functions as well.
As for "müsait," it can mean "convenient" as in "suitable," but it can also mean "favorable," "genial," and "opportune."
TDK also has an online dictionary, but it is in Turkish, so I had to rely on Google Translate to try to understand it. Knowing that not everyone appreciates Google's translations, I won't repeat, verbatim, what was translated, but it did offer slightly different nuances to both of these two words. If Google's translation was correct, however, it appears that TDK considers the primary meaning of "mevcut" to be "available" as in existing and associates "convenient" with "müsait."
Having said all of that, I also went to Tatoeba, which was enlightening. A search for "mevcut" produced 87 results (https://goo.gl/PqlP8Q) and 34 (https://goo.gl/LcRwQo) for "müsait." Nearly half of the translations for "mevcut" were tied to the meaning of "present" as in "All are present." About a third were tied to the meaning of "available" as in "Is there a room available for tonight?" Just a tenth were tied exclusively to "existing" as in "___ does not exist," but many of these were idiomatic.
A couple of times, I saw "mevcut" used exactly as we've been taught to use "müsait" -- with "not" (değil) to express unavailability of a person, and "müsait" used to express availability (or lack thereof) of a room so now I wonder if these two words are so close in meaning that many use them interchangeably. Having said that, when I took a closer look at the 34 results returned for "müsait," nearly all of them translated the word as "available" and most of the time these translations were tied to the availability of a person (although about a fifth were tied the availability of a "thing" such as rooms and seats).
Observation of Tatoeba translations is far from a perfect science for a number of factors. Translations could be inaccurate and submissions to the database may not reflect the actual quantities of usage in real life, but, at the very least, it gives one a better sense of how a word or phrase might be used.
If nothing else, I hope this gave you a better "feel" for these two words. It might also be helpful to know that "mevcut" appears to be a much more commonly used word than "müsait." Searching for just pages in the Turkish language, Google returned nearly 48 million results for "mevcut" and just a bit more than 4 million for "müsait."