Hi, I'm trying to understand exactly what the Laplace distribution is used for. Currently, I'm not seeing much real world application for it. Is anyone familiar with its importance?
Did you look at Wikipedia?
It says "The Laplacian distribution has been used in speech recognition to model priors on DFT coefficients  and in JPEG image compression to model AC coefficients  generated by a DCT.
The addition of noise drawn from a Laplacian distribution, with scaling parameter appropriate to a function's sensitivity, to the output of a statistical database query is the most common means to provide differential privacy in statistical databases.
In regression analysis, the least absolute deviations estimate arises as the maximum likelihood estimate if the errors have a Laplace distribution.
The Lasso can be thought of as a bayesian regression with a Laplacian prior."