The standard lasso uses an L1 regularisation penalty to achieve sparsity in regression. Note that this is also known as Basis Pursuit (Chen & Donoho, 1994).
In the Bayesian framework, the choice of regulariser is analogous to the choice of prior over the weights. If a Gaussian prior is used, then the Maximum a Posteriori (MAP) solution will be the same as if an L2 penalty was used. Whilst not directly equivalent, the Laplace prior (which is sharply peaked around zero, unlike the Gaussian which is smooth around zero), produces the same shrinkage effect to the L1 penalty. Park & Casella (2008) describes the Bayesian Lasso.
In fact, when you place a Laplace prior over the parameters, the MAP solution should be identical (not merely similar) to regularization with the L1 penalty and the Laplace prior will produce an identical shrinkage effect to the L1 penalty. However, due to either approximations in the Bayesian inference procedure, or other numerical issues, solutions may not actually be identical.
In most cases, the results produced by both methods will be very similar. Depending on the optimisation method and whether approximations are used, the standard lasso will probably be more efficient to compute than the Bayesian version. The Bayesian automatically produces interval estimates for all of the parameters, including the error variance, if these are required.
Chen, S., & Donoho, D. (1994). Basis pursuit. In Proceedings of 1994 28th Asilomar Conference on Signals, Systems and Computers (Vol. 1, pp. 41-44). IEEE. https://doi.org/10.1109/ACSSC.1994.471413
Park, T., & Casella, G. (2008). The bayesian lasso. Journal of the American Statistical Association, 103(482), 681-686. https://doi.org/10.1198/016214508000000337