I know they are two different approximation approaches to explicit models(which require approximation, that is transforming a non-optimization problem to an optimization problem to avoid the intractability of the marginal likelihood), how can I draw a clear line between the two? The Markov chain approximations, for example Pagerank algorithm and the Restricted Boltzmann Machines, and the variational approximations, like LDA, seem alike to each other and all for approximating the hidden variables of some functions. 

Absolutely I am wrong. Could anyone please help differentiate these two approaches? 

I learned that the former uses the stochastic approximations(usually Markov Chain Monte Carlo) while the later utilizes the deterministic approximations(always variational methods). The MCMC allows direct optimization of the variational lower bound. But still feel frustrated to intuit the differences.