Equilateral does not imply equiangular. A square can be ‘leaned over’ to make a rhombus with two distinct angles. Likewise a regular hexagon can be squashed to produce unequal angles while maintaining equal sides.
Equiangular does not imply equilateral. A square can be stretched into a rectangle, & a regular hexagon can be stretched along a pair of opposite sides, while remaining equiangular.
A regular polygon is both equilateral and equiangular.