A Gaussian Process Regression is a non-parametric model, which means it heavily depends of the training data you use. This means that, in this case, every training point will have a coefficient of it's own for the covariance/kernel function, which makes a comparison between Gaussian Process coefficients and Polynomial Coefficients impractical. For a better understanding of the method I would strongely suggest the following online book http://www.gaussianprocess.org/gpml/, specially the chapter 2 which focus on regression.
If you want to compare both methods my main recomendation is to used cross-validation. The simpliest way to do it is to divide your training data in two sets, a training and a validation one. You can then train both models in the training set and evaluate the loss in the test set. A Gaussian process is able to represent any dataset with 0 error depeding of your parameters, thus a cross validation is needed to really be able know if the model is good enough to be generalized.
