They are synonyms. The use of each verb varies by country and context. "Escoger" can only be used when there are several options to choose from, while "elegir" can be used to indicate a decision in general, and does not necessarily carry the connotation of selecting one option from several already determined.

Aunque en la actualidad se consideran sinónimos, en el pasado así no lo era, tenían matices diferentes. Y en Latinoamérica se prefiere el verbo elegir por sobre escoger.

Not much difference, but some countries may have a preference.