In my experience the two terms seem interchangeable, but undoubtedly there is a difference between a family's crest and its coat of arms. What is it?
There is no such thing as a family crest nor a family coat-of-arms.
A crest is the device the device/object attached to the top of the helm (helmet). The coat-of-arms is the armiger's device portrayed upon the shield.
The crest is one component of a coat-of-arms, which can be used as a simplified symbol when the full coat-of-arms is too detailed e.g. on engraved cutlery.
Confusion between the priviledge or right to have a coat-of-arms and the actual arms has lead to an understandable wide-spread misapprehension. The actual coat-of-arms is the personal property of the holder.
The right to bear arms is heritable, so the sons and in some circumstances, daughters of the person who has had arms granted to them, can also use a coat-of-arms. However, only one person owns a particular coat-of-arms, so during his lifetime, sons use a slightly different version of the arms. Such differenced arms have extra charges added to the shield, the colours are changed, or some other modification is made. Only rarely is the crest part of the coat of arms modified, so all the sons have different coats of arms that include the same crest. In England at least, no crest has ever been granted that was not part of a coat of arms.
The rules for differencing are complex and each country that grants arms has different rules.
In Scottish Heraldry, the concept exists of a crest badge - "As a mark of allegiance to their chief, members of a clan are permitted to wear a clansmen's badge, consisting of their chief's crest surrounded by a strap and buckle device on which the chief's motto is inscribed" (See Wikipedia).
As the right to have a coat-of-arms is associated with power and influence, there is a long history of mis-use. Commercial companies promote the idea of a family crest or coat-of-arms to sell merchandise depicting coats-of-arms. For discussions on this topic, see Dick Eastman's Online Genealogy Newsletter, e.g. this posting).
From the Oxford English Dictionary:
Coat of Arms
Heraldic insignia or devices. coat armour = ‘coat of arms,’ originally a vest of silk or other rich material embroidered in colours, worn over the armour of a knight, to distinguish him in the lists or on the field of battle.
Heraldry. A figure or device (originally borne by a knight on his helmet) placed on a wreath, coronet, or chapeau, and borne above the shield and helmet in a coat of arms; also used separately, as a cognizance, upon articles of personal property, as a seal, plate, note-paper, etc.
Coat of Arms
A coat of arms is a unique heraldic design on a shield or escutcheon or on a surcoat or tabard used to cover and protect armour and to identify the wearer.
A crest is a component of an heraldic display, so called because it stands on top of a helmet, as the crest of a jay stands on the bird's head.
A crest is only part of the coat of arms. See this Wikipedia page for the full list of the parts of a coat or arms.
The College of Arms in London, which is a branch of the Royal Household, is the official repository of coats of arms (for English, Welsh, Northern Irish and Commonwealth families). They have an FAQ section on their site, which includes a colored illustration of crests without their shields. https://www.college-of-arms.gov.uk/resources/faqs
The College of Arms states that "a crest is a specific part of a full achievement of arms: the three-dimensional object placed on top of the helm."
Often when people had jewelry or silverware engraved, they would only use the family crest, as the entire coat-of-arms was very detailed, and would be too small to make out, or too expensive to engrave.
Although there is no such thing as a "family" coat-of-arms, there are family crests. A man can passed down his coat-of-arms, and his sons and grandsons will "difference" the CHARGE (the symbols on the shield), but usually the CREST (the portion above the helmet) remains the same for the entire clan.
Crest is an object placed above helm usually on a torse or crown (for nobility). A family shares the same crest in most of Europe.
Coat of arms is what is on the shield (and what a knight also had on his coat covering his armor, thus the name) In England (and maybe some other parts of Europe) a Coat of arms is personal and could be handed down to the main heir only after death. No two people could bear the same coat of arms at the same time.
In Switzerland (where I am from and where nobility was banned when it was founded) as well as Germany however, a family usually shares a coat of arms (the literal translation of "coat of arms" from German is "family blazon") In Switzerland crests were not regulated and seldom used (since there was no nobility), sometimes being shared by the family, sometimes personal.
Often coat of arms is falsely used to describe the full achievent (crest, torse/crown, helm, manteling, shield, and sometimes supporters and ribbon) however this is incorrect in the strict sense of heraldry. Crest is strictly speaking not part of the coat of arms (but often used so), but both are part of the achievement.
In general it can be said that different norms were used depending on time and place.