L1 drives a weight to exactly zero; L2 never does.
w₁ , w₂
The penalised solution sits where a loss contour just touches the L1 diamond or the L2 circle through it. The diamond has corners on the axes, so as λ grows the L1 path runs into an axis and w₂ becomes exactly zero; the circle has none, so L2 pulls the weights towards the origin without zeroing either. The loss is illustrative.