They look to both be almost exactly the same size- 3.9 inches in diameter.
Echo Dot: 100 x 100 x 89 mm HomePod Mini: 98 x 98 x 84mm
Echo Dot: 100 x 100 x 89 mm HomePod Mini: 98 x 98 x 84mm
However, there's probably a bit of a difference with the amount of on board processing that the two devices can do.
For example, my iPhone can do voice to text while on airplane mode.
{guessing} Similar on board processing would allow a HomePod Mini to work with HomeKit without needing to go to the cloud - Amazon working with Hue does need to go out to the cloud to get that voice to text to command layer. {/guessing}