The more MATLAB-like solution:
Animate[
Show[
ParametricPlot[{Sin[4 x] Cos[x], Sin[4 x] Sin[x]}, {x, 0, y},
PlotRange -> {{-1, 1}, {-1, 1}}],
ParametricPlot[{Sin[4 z] Cos[z], Sin[4 z] Sin[z]},
{z, y - 0.02 \[Pi], y}, PlotStyle -> Directive[Red, Thick]],
Graphics[{Red, AbsolutePointSize[8],
Point[{Sin[4 y] Cos[y], Sin[4 y] Sin[y]}]}]],
{y, 0, 2 \[Pi],0.001 \[Pi]}
]
Here's the gif output:

Personally I prefer that the full orbital be kept as background with comet moving on it. In that case I change the parameter maximum to 2[Pi] instead of y, in the first ParametricPlot. (I use angular frequency 4 instead of 12 to make the animation smoother. If you raise the angular frequency, you should use more points, that is, smaller step, in animation so that it is smooth enough. Otherwise it looks silly to use comet plot.)