In general chemistry, we learn that Lewis bases have an electron pair that they donate to Lewis acids, which accept electron pairs. In organic chemistry, we learn that nucleophiles donate electron pairs to electrophiles, which accept electron pairs. So what's the difference between acid-base chemistry and nucleophile-electrophile chemistry?
Nucleophilicity describes the rates of reaction; that is, a molecule that is more nucleophilic will add faster to an electrophile than a molecule that is less nucleophilic would. Similarly, electrophilicity has to do with how fast nucleophiles would react with an electrophile. A better electrophile would accept a nucleophile faster than a poor electrophile would.
Lewis basicity (or just simply, "basicity"), on the other hand, talks about how stable the bond that forms is. Thus, basicity is fundamentally about acid-base equilibria, and not about rates of reaction. A stronger base forms a stronger bond to a Lewis acid compared to the bond a weaker base would form with the same acid.
In terms of a reaction coordinate diagram, the reaction barrier, or the "hump", is related to the rate at which a reaction proceeds. Since rate corresponds to nucleophilicity, we can say that the size of the reaction barrier corresponds to the nucleophilicity of a molecule. Similarly, the difference in energy between the start and the end is related to how exergonic a reaction is, and so it describes the equilibrium between the two states (start and end) and how stable one state is over the other. Thus, the difference in energy, which is known as the free energy of reaction, corresponds to the basicity. You can see this all illustrated in the figure below.

As an example, let's look at the following reaction between methyl iodide and 8-amino octane-8-thiol. The electrophile will be the methyl iodide, because the C-I antibond can accept electrons. The nucleophile will be the 8-amino octane-8-thiol, but which end will be the nucleophilic center? We have a choice between sulfur and nitrogen, as both have lone pairs that can donate to the electrophile. (continued below figure)

(Step 1) Sulfur is more nucleophilic than nitrogen, even though nitrogen is more basic than sulfur. Since nucleophilicity is about rates, that means that sulfur "gets to" the electrophile and reacts with it faster than the nitrogen does. Thus, sulfur attacks the methyl iodide and forms a bond first.
(Step 2) Now we have sulfur bonded to the methyl group, and the fact that the nitrogen is more basic comes into play. Although nitrogen attacks the methyl group slower than sulfur does, a bond between sulfur and carbon is weaker than the bond between nitrogen and carbon (because nitrogen is more basic). Thus, the nitrogen can kick the sulfur off the methyl group, and this new molecule is much more stable than the previous one, and so its formation is favored in equilibrium conditions.
(Step 3) Finally, the free iodine deprotonates the positively charged nitrogen to make everything neutral in charge.
Thus, better nucleophiles (and electrophiles) react first. Stronger bases (and acids) are more favorable to form in equilibrium. The key thing to note is that if the reactions are irreversible (this could happen if the temperature were really low), this means that the molecules don't have a chance to find the most energetically stable formation, and so they stay in whatever shape they form first. This means that under irreversible conditions, nucleophiles determine what the products are.
Nucleophilicity and electrophilicity are kinetic descriptions. Basicity and acidity are thermodynamic descriptions. Under irreversible (i.e. kinetic) conditions, stronger nucleophiles and stronger electrophiles form the products. Under reversible, equilibrium (i.e. thermodynamic) conditions, stronger bases and acids form the products, although at first, the nucleophiles might react to form an intermediate species.
Thus, although nucleophiles are usually bases and bases are usually nuclephiles, the description of what makes a better base or a better nucleophile is an important and real distinction that has to be made in order to understand why things happen the way they do. In addition, of course, the exact same contrast exists between electrophiles and acids. For an interesting biochemical application of the difference between nucleophilicity and basicity, look up "Kent coupling" on Google
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