On Aperiodic Packing - and Covering

In continuation of this this earlier post here:

a simple question:
"Are there convex 2D shapes which cannot tile the plane and for which the best packing fraction is achieved by an aperiodic arrangment and not by a lattice one?" (the question can be posed for general -non-convex shapes as well).

If such regions exist, which among them shows max difference in packing fraction between its best lattice and best aperiodic pack?

Mathworld says: "Gauss proved that the hexagonal lattice is the densest plane lattice packing with unit circles and in 1940, L. Fejes Tóth proved that the hexagonal lattice is indeed the densest of all possible plane packings."

Fejes Toth has also proved that a lattice arrangement is the best packing for any centrally symmetric shape. So one has to look beyond such shapes for answers to our question.
And just as aperiodic packing, one can ask about (convex and general) shapes (in 2D and beyond) for which aperiodic covering beats lattice covering.