Besides of the similar name there is actually no close relation between the algorithms. Label propagation is a simple algorithm described here: Zhu, X., & Ghahramani, Z. (2002): Learning from labeled and unlabeled data with label propagation. It is a semi supervised algorithm to try and predict missing labels based on graph structure.
Belief propagation is a family of algorithms which runs on top of probabilistic graphical models. There are many variants including variants for discrete and continuous distributions, factor graphs, directed and undirected graphs, etc. the goal is to infer hidden state of the unobserved variables. The max product variant tries to infer the state (argmax) which maximizes some probability function, while the sum product variant tries to infer the marginal probability. I suggest taking the pgm course in coursera.
There is no direct way to compare those algorithms or to say which will work better. It is true that both can use for finding the best assignment of those missing values.
In GraphLab Create we have label propagation implemented and belief propagation is on our roadmap.