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Environmental variations can significantly influence how populations compete for resources, and hence shape their evolution. Here, we study population dynamics subject to a fluctuating environment modeled by a varying carrying capacity changing continuously in time according to either binary random switches, or by being driven by a noise of continuous range. We focus on a prototypical example of two competing strains, one growing slightly slower than the other. By systematically comparing the effect of a binary versus continuously varying environment, we study how different noise statistics (mean, variance) influence the population size and fixation properties. We show that the slow strain fixation probability can be greatly enhanced for a continuously varying environment compared to binary switches, even when the first two moments of the carrying capacity coincide.