I am solving the Lasso problem
minimize 1/2*|| Ax - b ||_2^2 + lambda || x ||_1
with CVX and getting good results for my study but when I apply ADMM algorithm I am getting very different results. Why is there such a difference? I would like to understand the solution of the problem better but cant follow what CVX does.
Answers and suggestions will be very helpful.

Is the optimal objective value the same, within a reasonable tolerance? If so, perhaps the argnin is no unique. Does CVX report the problem is solved to optimality? How about ADMM? You have not shown us any code or any example or results, so how can we provide reasonable explanations?