Simply, langoustines are much smaller and a portion is several. A langouste is large; one would be big enough for 1--2 people.
Other names for langoustines include (from wikipedia):
Nephrops norvegicus, Norway lobster, Dublin Bay prawn, or scampi
In French, a longer version of the name is Langoustine commune (again wikipedia):
Contrairement à ce que son nom peut suggérer, la langoustine n’est pas une petite langouste
Despite what its name may suggest, a langoustine is not a small langouste
While they can grow to 25 cm long they're normally served about half to 2/3 that size. As typically presented, they're rather like large straight shell-on prawns, though a special long thin fork may be used to extract the meat.
Langouste (again from wikipedia) is known as:
Palinurus elephas, European spiny lobster, common spiny lobster, or Mediterranean lobster
In the parts of France I've most often visited, this is less common than the other major lobster, which is called homard, homard européen, homard breton* or scientifically Homarus gammarus.
For those who can muddle through a bit of French, here's some further reading (the last paragraph is particularly relevant): Quelle différence entre un homard et une langouste ?.
* Breton as in Brittany, which is where I tend to go.