Degree vs Valence of a vertex in a graph To my understanding, "degree" and "valence" mean the same thing: the number of edges incident to a vertex (including multiplicity for loops).
Is there a difference in the contexts in which they are used?
 A: Both terms are equivalent. However, the term "degree" is used more often these days. In old publications, such as the ones of Erdős (c.f. "On a valence problem in extremal graph theory", Erdős and Simonovits, 1972), the term valence is ocurring more frequently.
In today's graph theoretical literature, however, it has almost disappeared (standard works such as Bondy's and Murty's "Graph Theory", already no longer mention it, and it is not appearing in their older work called "Graph Theory with applications" (1972), either). Diestel's "Graph Theory" uses both terms equivalently ("The degree (or valency) [...] of a vertex $v$ is the number of edges at $v$", p.14). However, he mostly uses the term "degree".
In chemical graph theory, one often tries to strictly separate the terms in order to make a clear distinction between the valence of chemical bonds and an abstract graph theoretic model (see for example "A review on molecular topology: applying graph theory to drug discovery and design" by Amigó et. al.).
Whether there are other applications in which the term "valence" takes on a specifically different meaning than the term "degree", I can't say for sure. In pure graph theory it does not.
A: Valence also has a meaning in complex dynamics -- the number of K-rays that land on each point of a periodic orbit (found in Milnor's Periodic Orbits, External Rays, and the Mandelbrot Set: An Expository Account).
