We study the routing policy choice problems in a stochastic time-dependent (STD) network. A routing policy is defined as a decision rule applied at the end of each link that maps the realized traffic condition to the decision on the link to take next. Two types of routing policy choice models are formulated with perfect online information (POI): recursive logit model and non-recursive logit model. In the non-recursive model, a choice set of routing policies between an origin-destination (OD) pair is generated, and a probabilistic choice is modeled at the origin, while the choice of the next link at each link is a deterministic execution of the chosen routing policy. In the recursive model, the probabilistic choice of the next link is modeled at each link, following the framework of dynamic discrete choice models. The two models are further compared in terms of computational efficiency in estimation and prediction, and flexibility in systematic utility specification and modeling correlation.