Fulvous Hawk & Death's Head Hawkmoth have very similar caterpillars. So much so that I have always wondered why they're not in the same genus...

Anyhoo, here are some illustrations to help ID these gorgeous beasties.

Like many species, the caterpillars have GREEN and BROWN forms. It would seem from records on iNat that the green forms are much more common (or perhaps more easily seen?) than the brown forms. In all forms Acherontia has a stubby, bobbly tail, and Coelonia has a longer, granular tail.

Desriptions and illustrations here show late instar larvae. They can look pretty different in early instars.

GREEN FORMS

Image credits:
Acherontia atropos - @rjpretor
Coelonia fulvinotata - @leani

Acherontia

can be many shades of green or yellow (see image below) with stripes of green and/or blue and speckles.
no dark line down the centre of the back.
tail stubby, usually floppy (can be erect in earlier instars) and covered in bobbles ('pimples').
spiracles usually readily visible

Coelonia

shades of green, stripes of blue or purplish, no speckles.
wide dark line down the back - purplish or blue, or can be a sort of pale beige-orange.
tail longer, usually held at an angle, can be sort of granular but not bobbly.
spiracles often not visible

BROWN FORMS

Image credits:
Acherontia atropos - @craigmain
Coelonia fulvinotata - @suncana

Image credits:
Acherontia atropos - @nick-de-swardt
Coelonia fulvinotata - @suemarsden

Acherontia

fairly uniform pale beige-brown heavily spotted with white dots.
diamond pattern along the back.
tail stubby, usually floppy (can be erect in earlier instars) and covered in bobbles ('pimples').
thorax (which looks like the top of the 'head') is smooth, whitish, with brown checkerboard pattern.

Coelonia

fairly uniform darkish brown on top and paler on the sides, with dark brown patch (usually) on the anal prolegs.
no diamond pattern on the back
tail longer, usually held at an angle, can be sort of granular but not bobbly.
thorax (which looks like the top of the 'head') has lovely spiny ornamentation, with brown checkerboard pattern.