In contrast to fixed-effects regression models, latent growth models are random effects models that allow for inter-individual differences in the intercept and growth rate (individuals can have different trajectories). Latent growth models also take random measurement error into account and separate true change from fluctuations due to random noise.

As Christian Geiser says, the random effects allow differences in growth. This growth can be linear or polynomial (or whatever), and you can decide which of the contrasts to be random. So it is not a polynomial OR random/latent decision. Having several random variables can lead to computational issues, but there aren't many processing that correspond with a high degree polynomial. How high a degree is your model?

Polynomials will break down for extrapolations.

The logistic model gives meaningful estimates of the initial (unhindered/maximal) growth rate, the upper abundance limit (ecological capacity?) and the inflection point ("EC50").

The logistic model allows to estimate effects of treatments/environmental factors on these (relevant) parameters.

If you just want a curve that describes the relation between x and y, better use splines (natural splines)! They are more flexible, flexibility can be adjusted by cross-validation, and they have a more natural behaviour, particularily at the limits of x. Using polynomial for this purpose is medival.

Polynomials are not reflecting any underlying physical/biological law. They are just stupid "passe-partout" functions that can serve to smooth or interpolate data. They don't explain anything.

I have examples of growth models in my software "FittingKVdm". You can try it and let me know your experiences/suggestions.

www.lerenisplezant.be/fitting.htm