davelikestoplay

As soon you add the shadow (as you did for Mars) you will directly achieve a more 3-dimensional impression. Besides a slightly bending of the rings, as suggested by @Ron P., which works like changing the focal lengths of the virtual camera.

If you really want to do this as an all vector project in AD, you could start with a circle filled with a background color that approximates the average color of the center band, probably a light, slightly off-white with a bluish tint. Then draw several rectangles for the other bands, clipped to the circle, convert each one to a curve & use the node tool to slightly bend the top & bottom edges of each one to approximate the roundness of the sphere, & fill each one with a color that approximates the average color of that band.
The turbulence of the upper brownish band is going to be hard to duplicate but you could tediously add multiple smooth nodes & offset them one or a few at a time to create that effect.
It is doable but you could spend hours refining it & breaking the bands up into smaller & smaller parts to get more color variation, & still end up with something that looks artificial, blotchy, & not very recognizable as Jupiter.

Affinity Designer doesn’t have any vector warping capabilities yet, even if it had photos warping tools, the objects would be rasterised.
The only way to get the spherical effect is to manually apply bend or draw large circles to simulate the arcs and clip within the ellipse.

It hardly seems worth the effort for such small sizes and the complexity of the subject, you could make a kind of cut down version using the  symbolic parts such as the eye storm and the banding.
 

Is there a specific reason/advantage to create this as vectors? Especially for tiny output sizes?
Or is your question rather how to reduce the original's details to appear less detailed but more graphic and abstract?

Hi @davelikestoplay,
Welcome to Affinity Forums
I agree with @firstdefense. Even the larger size is too small to discern any meaningful detail other than a few bands of colour and an orange ellipse. One way to have a good reference of the detail needed to start is simply reducing the raster image to the intended final size to check it (or even use it directly as a final image without vector tracing). Another is to squint your eyes to suppress superfluous detail, focus the main components that define the image then draw them. You will probably end up with a stylized/simplified version that people may not recognize/interpret as a planed if they aren't aware of Jupiter's appearance/details.
Just for reference (72 px):
 

@davelikestoplay,
Apply the Live Spherical Filter to get the bending or your planet to look like a ball. You can find it in the Layer Menu>New Live Filter Layer>Distort or I just use Live Filter icons at the bottom of the Layer Studio, it looks like an hour-glass.