What is the difference between nucleophilic substitution and electrophilic addition?

It's really in the terms "substitution" and "addition" that we find a meaningful difference.

NUCLEOPHILIC SUBSTITUTION

Nucleophilic substitution is when a nucleophile attacks an electrophilic site (i.e. a particularly electropositive site) and displaces a substituent in order to form a new molecule.

One example of such a reaction looks something like this (without consideration of #"S"_N1# vs. #"S"_N2#, for simplicity):

We can see that #"NH"_3# displaces #"Br"^(-)#, because #"NH"_3# is a strong lewis base (nucleophile), while #"Br"^(-)# is a good leaving group.

We can conclude that nucleophilic substitution reactions will overall have had a leaving group leave from the substrate, due to the participation of a nucleophile.

ELECTROPHILIC ADDITION

Electrophilic addition is addition onto an electrophile without displacement.

One common electrophilic addition reaction is actually the halogenation of alkenes to generate vicinal dihalides. It's the dihalogen that acts as the electrophile.

Initially the #"Br"_2# is actually not polarized. But when it approaches the alkene, since bromine is more electronegative than carbon, it slightly draws electron density from carbon, which then is led to donate electrons into bromine's antibonding orbital.

That causes a London-Dispersion distortion in the electron cloud of #"Br"_2#, generating an induced dipole which, along with bromine donating electrons back to the other carbon's antibonding orbital, facilitates the movement of electrons into the bonding orbital of the rear bromine and breaks the bond.

The bromonium intermediate is unstable, allowing #"Br"^(-)# to backside-attack and generate the vicinal dihalide, giving the anti addition product.

For this, we can say that electrophilic addition reactions will overall add something onto a previously electron-rich molecule, but the net result does not involve something having left the original substrate by the time the reaction is over.