Continuous response variables are often transformed to meet modeling assumptions, but the choice of the transformation can be challenging. Two transformation models have recently been proposed: semiparametric cumulative probability models (CPMs) and parametric most likely transformation models (MLTs). Both approaches model the cumulative distribution function and require specifying a link function, which implicitly assumes that the responses follow a known distribution after some monotonic transformation. However, the two approaches estimate the transformation differently. With CPMs, an ordinal regression model is fit, which essentially treats each continuous response as a unique category and therefore nonparametrically estimates the transformation; CPMs are semiparametric linear transformation models. In contrast, with MLTs, the transformation is parameterized using flexible basis functions. Conditional expectations and quantiles are readily derived from both methods on the response variable's original scale. We compare the two methods with extensive simulations. We find that both methods generally have good performance with moderate and large sample sizes. MLTs slightly outperformed CPMs in small sample sizes under correct models. CPMs tended to be somewhat more robust to model misspecification and outcome rounding. Except in the simplest situations, both methods outperform basic transformation approaches commonly used in practice. We apply both methods to an HIV biomarker study.