We investigate two inherently different classes of probability density functions (pdfs) that share the common property of power law tails: the α-stable Lévy process and the linear Markov diffusion process with additive and multiplicative Gaussian noise. Dynamical processes described by these distributions cannot be uniquely identified as belonging to one or the other class either by diverging variance due to power-law tails in the pdf or by the possible existence of skew. However, there are distinguishing features that may be found in sufficiently well sampled time series. We examine these features and discuss how they may guide the development of proper approximations to equations of motion underlying dynamical systems. An additional result of this research was the identification of a variable describing the relative importance of the multiplicative and independent additive noise forcing in our linear Markov process. The distribution of this variable is generally skewed, depending on the level of correlation between the additive and multiplicative noise.