Please note that it is specifically Lewis basicity, not any other definition of basicity.
Consider the interesting trend in nucleophilicity and basicity of the fluoride and iodide anions. The reason claimed for the weak nucleophilicity of the fluoride ion in polar protic solvents is due to the hydration effects which it experiences as it interacts with solvent molecules. This hydration effect effectively prevents it from reacting with electrophiles. In a sense, the hydration effect severely decreases the availability of the electron pair for donation.
Now, although it is claimed to be a weak nucleophile in aqueous solution, the fluoride ion still behaves as a strong Lewis base. This is logical because the interacting species are different in the two different situations. In the case of nucleophilicty, we are considering its nucleophilicity of the species towards another electrophile (e.g. a haloalkane) placed in solution.
In the case of basicity, we are considering it, in terms of its hydrolysis reactions with the water molecules in the hydration shell, producing the hydroxide ions. Thus, the solution becomes an alkaline one and we say that it is basic.
Now, if we use the same perspective to view nucleophilicity, we can thus say that the water molecule is the electrophile and that the fluoride ion is indeed a strong nucleophile because it has managed to attack the electron-deficient hydrogen atoms on the water molecule.
Consider this: What if I remove all the hydroxide ions produce from the hydrolysis of water from an aqueous solution of fluoride ions and add in an acidic species? Would we still observe a reaction between the fluoride ions and the acid species?
My point is that when we consider Lewis basicity and nucleophilicity of reacting species we are not making a fair comparison and thus, we say that fluoride ions are strong bases but weak nucleophiles in polar protic solvents. This may be true based on the Arrhenius definition of acids and bases. But if we were to look purely at Lewis basicity, it would be certainly be inaccurate.
From the perspective of interactions with water molecules, the fluoride ions are definitely good Lewis bases and also good nucleophiles. From the perspective of interactions with other reacting electrophiles placed into solution, the fluoride ions would be both weak Lewis bases and weak nucleophiles, neglecting the hydroxide ions (because even though the acidic species may react with the hydroxide ions, it is not reacting with the fluoride ions, thus we would see those reactions of the acids with the hydroxide ions as a side reaction).
Therefore, is it appropriate to equate these two terms because they both refer to the electron-donating ability of a chemical species?
If I have overlooked something (i.e. some other differences between the two terms), please inform me via the comments.