Being French I tend to refer to abv more than proof when discussing the alcohol content of a drink. Proof appears from time to time through. But what is the difference ?

Both abv. and proof refers to the measurement of the content of ethanol in a liquid. The statement of the measurement is compulsory by law in many countries. That already make two common points.
What are the differences?
– the history
– the calculation

What is “proof”?
For centuries (pre 20th century actually), liquor was “proofed” at the distillery by adding gunpowder and lighting it on fire. If it didn’t light, the alcohol content was too weak. If it burned yellow, too strong. If it burned blue, the proof was just right (that was around 57 percent, or 114 proof). A century ago, the US established a standard according to which quality spirits were “bonded” at 100 proof ( 50 % abv.). Over time, federal and state excise taxes on higher-proof spirits drove down the average proof, as did health concerns and consumer preference. Please note that  the “proof” systems are not the same in the UK as the USPlease note that “Overproof” or “Navy Strength” means the spirit—usually rum or gin—has an ABV of 57% or higher.
Do you need high proof for a spirit to taste better? No. Or more precisely it depends on the spirit you want to make. For example, in the case of absinthe, you have to bottle it at high proof because of the botanicals. If the abv. isn’t high enough, the compounds will deteriorate. The spirit becomes hazy with sediment, which will not look nice.

What is “abv.” ?
It means alcohol by volume. The International Organization of Legal Metrology is in charge to recommend how to measure alcohol strength. It is defined as the number of millilitres (mL) of pure ethanol in 100 mL of solution at 20 °C (68 °F). The number of millilitres of pure ethanol is the mass of the ethanol divided by its density at 20 °C, which is 0.78924 g/mL. The abv standard is used worldwide.
However, in France it is underwood differently. In France, alcohol content is in degrees Gay-Lussac (GL). A technician tests the solution with a hydrometer a hydrometer. Then the person expresses alcohol strength as parts of alcohol per 100 parts of the mixture. Thus, a spirit with 40% alcohol by volume equals 40 degrees GL. This said, most of what you see on the bottles in France is the standard abv.