Updated Feb 6
It isn’t an either/or type issue. Both factors are involved.
However, it is apparently sequential. The folding tends to occur in a certain order.
As I vaguely recall, the first secondary structure to appear is the alpha-helix (if there is one), and this can start forming at the N-terminus, as it is still coming out of the ribosome. Then, the beta-sheets form a bit later.
I expect that the formation of individual alpha-helices and beta-sheets would based on enthalpy, since they are held together with hydrogen bonds.
Later, the alpha-helices and beta-sheets need to be arranged in tertiary structure, in relation to each other, with certain areas facing certain other areas.
This stage of folding is mainly based on the hydrophobic effect, which is driven by entropy.
It is a bit counter-intuitive at first. When hydrophobic amino acids (Phe, Val, etc) are facing the water environment, this forces the surrounding water molecules to become more ordered into a “cage”. That means lower entropy.
When those hydrophobic side chains are folded inwards, to the core of the protein, this allows the water molecules to become more free and entropic. That increase in the water entropy is larger than the decrease in the protein entropy.
Then, you may get the sort of mixed effect of an alpha-helix or a beta-barrel that is hydrophobic on one side, and hydrophilic on the other. This arrangement is used in membrane channels.
My understanding is that, early formation of these large entropy-driven structures then brings together smaller areas that can later form enthalpy-based hydrogen bonds, salt-bridges, disulfide bonds, and interactions with co-factors (e.g. a metal).
So, you need both forces, acting at different times, on different levels of the folding process.
As always, there is more complexity around the corner. If there are chaperones involved in the folding process, it seems to be mainly about controlling the hydrophobic vs hydrophilic exposure.
Also, with a protein that is normally associated with a membrane, the final stage of the folding process may need to occur when it is in that lipid environment (so that the hydrophobic side chains can point outwards).
Disclaimer: Thermodynamics and chemical energetics are among my least-favourite areas of science, so there are certainly people who know much more than I do about them.