Classic wins the all-purpose case; Chef Collection wins when maneuverability is the point.
The Classic is the one to choose when you want a single 12-inch pan to sear proteins, shallow fry, make cornbread, bake a skillet meal, and hold food with a deeper rim. Its published 2.25-inch cooking depth is only a quarter inch greater than the Chef Collection’s 2-inch depth, but shape matters more than the bare number: the Classic keeps the familiar taller-wall skillet format while the Chef Collection uses a more open profile.
The Chef Collection is not a lesser skillet. Lodge specifically describes it as chef-inspired, with ergonomic handles and spatula-friendly sidewalls. Its published 11.81-inch cooking-surface length, 9.12-inch flat bottom, and 12.79-inch overall width are close to the Classic, so it does not ask you to give up the useful footprint of a 12-inch pan. It asks you to trade a little depth for easier access at the edge.
For most shoppers, the Classic is the cleaner recommendation because its shape is more flexible when the meal changes from eggs to a saucy braise to a baked side. Pick the Chef Collection if you already know your routine leans toward flipping, turning, sauteing, and sliding a spatula under food. Do not choose based on a live price or a retailer bundle: those can move, and the linked Chef Collection ASIN uses an Everyday Pan label that should be checked for included accessories.