Resonance structures are a more complete description of chemical bonding than are single Lewis structures because they allow us to show the electronic dynamics of a molecule with static pictures. This is not always necessary of course, as no one will complain if you draw one structure for ethanol but not all molecules are as simple as ethanol.
To get away from aromatic systems, take the difference in acidity of methanol and acetic acid. The only difference between the two molecules is a couple of electrons and an oxygen atom. The pKa of acetic acid is more than 10 units lower than the pKa of methanol. Can you explain this behavior? Induction, you say? Well that effect is not so strong, compare for instance acetic acid and chloroacetic acid. The pKa difference is about 2 units, a far cry from an ~11 unit difference between methanol and acetic acid. You can reason it out by showing that there are two equivalent resonance forms of the carboxylate anion, therefore you can see that the charge is evenly spread amongst two oxygen atoms. It is not the case that there is one neutral double bonded oxygen and one negative single bonded oxygen on the carbon atom. Its closer to reality to describe each oxygen with a one and a half bond and a half of a negative charge. With this spread in charge comes a stabilization of the anion and hence, a lower pKa (a stronger acid).
Resonance structures aid in understanding bond lengths. Take the example of the benzene molecule shown above. The bond length is listed as 140pm (see the picture above). A quick google search reveals a that carbon carbon single bonds have a bond length of 154pm whereas carbon carbon double bonds have a bond length of 134pm (http://www.science.uwaterloo.ca/~cchieh/cact/c120/bondel.html). We can see that the C-C bond length in benzene falls between that of a double and single C-C bond, without knowledge of Kekule structures/resonance structures this may seem puzzling. When you realize that the resonance structures are showing you that there are NOT three single bonds and three double bonds but closer to six one-and-a-half bonds, you begin to see the usefulness of resonance structures.
You can also consider the case of carbonyl addition reactions. Why is the carbonyl carbon an electrophile? Looking at any garden variety ketone/aldehyde, why should a nucleophile go and attach to the carbon and not the oxygen? Resonance structures help explain this behavior.
The stability of charges/radicals at allylic or benzylic positions are explained by resonance structures of the ions/radicals produced.
The list goes on and on. You will appreciate this more if you try and learn Organic Chemistry conceptually instead of by rote memorization. Pretty much 99% of first year O-Chem can be explained by 3 things: Sterics, Resonance, Induction.