[orgoclear]

As the title says,
What is the basic difference between nucleophilicity and basicity?
When does a strong base behave as a weak nucleophile?

[blind]

What defines nucleophilicity and basicity?  Nucleophilicity means how well (or "willing") a compound can attack an electrophile, which means sterics is involved.  Basicity is (and this is really simplified) how "willing" a compound accepts a proton.

An example of a strong base and weak nucleophile is lithium diisopropyl amine (LDA), used to generate enolates, but is not nucleophilic because of it's bulk.

[jj74]

Basicity is a termodinamic concept
Nucleophilicity is a kinetic concept

[bromonium]

Basicity is the ability of an atom or molecule to donate a lone pair of electrons to a proton (H⁺).  Nucleophilicity is the ability of an atom or molecule to donate a lone pair of electrons to carbon.

Within a period (row) on the periodic table basicity and nucleophilicity trends run in the same direction, and run opposite of electronegativity.  Moving from left to right elements become more electronegative but less basic and less nucleophilic.  When comparing two elements in the same period, the element further to the right is more electronegative and holds its electrons closer to its nucleus.  The less electronegative element is more "willing" to give up its electrons to form a bond, with either hydrogen or carbon.  Remember that nonbonding electrons usually lie closer to the atomic nucleus than bonding electrons, which are distributed between two atomic nuclei.  Consider the binary acids of some period II elements and their conjugate bases: CH₄/CH₃^-; NH₃/NH₂^-; H₂O/OH^-; HF/F^-.  HF is the strongest acid and therefore its conjugate, fluoride ion, is the weakest base.  Fluoride is also the weakest nucleophile.  Methyl anion is both the strongest base and the strongest nucleophile. Carbon is relatively electropositive and readily donates its lone pair to an electron deficient species to form a coordinate covalent bond; fluorine on the other hand is a very weak base and exists quite happily as the free fluoride anion in solution. OH^- is the prototypical base and is also a fairly good nucleophile, but it is quite stable compared to NH₂^- and the extremely unstable CH₃^-.

When comparing elements in the same group (column), size becomes more important than electronegativity in determining nucleophilicity.  Species which are more strongly attracted to solvent molecules have a decreased potential energy and are therefore less reactive. Smaller ions have a greater charge-to-surface ratio, or charge density, than larger ions. Fluoride and bromide ions both have a -1 charge, but this charge is distributed over a much larger area on Br^- than on F^-, so any given spot on the "surface" of F^- is more negatively charged than an equivalent spot on the "surface" of Br^-. Smaller ions are stabilized by their stronger attraction to polar solvent molecules, in particular they are more effective at hydrogen bonding with protic solvents than large ions are.  This stabilization makes them less reactive and weaker nucleophiles.

Regarding basicity and size, smaller atoms form stronger bonds to hydrogen than larger atoms; it takes less energy to break the H—I bond than it does to break the H—F bond. Of the halogens, iodide is the strongest nucleophile and the weakest base, and HI is the strongest hydrohalic acid (strong acids have weak conjugate bases). The same is true of SH^- and OH^-, where SH^- is a stronger/better/more reactive nucleophile and a weaker base.

The table in the attached image summarizes the trends in basicity and nucleophilicity.  Here's the URL to view it online:
http://img.photobucket.com/albums/v201/tupe_/periodictrends.png

My organic chemistry textbook briefly mentioned that, in aprotic (non-hydrogen bonding) systems, the nucleophilicity trend is "often" reversed.  Here the smaller nucleophile does not get the same potential energy decrease that H-bonding affords it, and the concentrated charge makes it more reactive.  If my memory serves me we did not focus on this much and most of the examples in class and on tests involved protic solvents.