What is the difference between "éplucher" and "peler"? It seems both terms are used for the same thing, e.g.
Éplucher une carotte
Peler une carotte
Although both words have a common early etymology in Classical Latin, they have long acquired different meanings and can rarely be used indifferently.
The etymology of peler is from latin pilare (action of removing the hair). In Old French, because of its proximity to pel (la peau, skin), the meaning broadened to mean "remove the skin", skin with hair like that of a fury animal but it began to be used for anything with a skin, including vegetables. So you will use peler for anything that has a skin: a rabbit, an apple, a banana, etc...
Éplucher comes from Old French espeluchier (itself derived from Vulgar Latin pilucare, from Classical Latin pilucare - "remove the skin") and was used not only to mean remove the skin, but remove the vermin, ie all non eatable parts of the animal.
Nowadays it is used for fruit and vegetables as well as for animals, and is used for the act of removing all non eatable parts of some food.
As far as culinary terms are concerned (both verbs can be used for non culinary purposes) éplucher is a far more deeper action that peler.
Peler is when you only remove the outer part of some food, sometimes you don't even have to use a knife but just to lift the skin and pull.
Éplucher means to remove the skin but also to remove all non eatable parts, therefore you will use a knife to get at parts that won't come off by themselves.
If it can be of any help to you, in English one usually uses "clean" for éplucher and "peel" or "skin" for peler.
To get back to your question: yes some people will use peler for carrots, but from my point of view it is highly debatable. I'm not sure it would be used in a professional kitchen since you don't want to serve parts that look bad. It might also depend on what type of carrots we are dealing with. The skin of young spring carrots is hardly blemished and can be eaten, whereas old fully grown carrots have a thick part on the outside that has to be removed.
Both verbs share the same meaning when applying to a fruit or a vegetable :
Éplucher une orange = Peler une orange
However, they have quite different acceptions when used in a figurative or slang way.
Eplucher un dossier.
Peler un dossier.(To examine a folder)
(slang/vulg) Se peler les couilles.
S'éplucher les couilles(freeze one's balls)