I think that both terms are close in a general sense. They belong both to "variation", that is the property to vary. Perhaps fluctuation may refer to variations in a long-term stationary/stable world, that is without any change of the mean/variance while variability is perhaps broader including stationary, but also non-stationary changes as abrupt thresholds and/or long-term linear or non-linear trend arising from instability or not-equilibria situations.
I appreciate the nice description Vincent has provided. But will like to add a bit more than the time scale, I guess fluctuation is a term that is used for a slightly unprecendeted variation while variations can be used for a more steady changes. In strict sense these are the terms which can easily be interchanged and used.
I think that fluctuation is used to describe sharp changes between extremes, while variability means that the phenomena is just changing and not kept steady.
For example, if the temperature is changed from 5 to 35 then 10 then 40 then -5 then 30, we can call it fluctuating, while if it changes from 15 to 20 to 25 then 21 then 19, ,,, we can call this variability, as the values are in the same average.
Mohammed that is what i meant when I said "slightly unprecendeted variations" for fluctuation and more steady change for variation.
https://eos.ucs.uri.edu/EOS_Linked_Documents/michu/michug11017.pdf
http://www.climatekelpie.com.au/understand-climate/climate-science/climate-variability-and-climate-change-whats-the-difference
https://www.igsoc.org/journal/57/203/t10J107.pdf
http://weather.unl.edu/RCM/IDB_Mexico/PDF/climatebasics.pdf
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