There is a difference. But before we go there, let's talk about the similarities to kind of get located.
Both lines are good value for the money. Both lines are discount copies of more expensive lines made by major manufacturers. Neither line will make a good cook a poor one, or vice versa. None of the pans are made from a single piece of stainless and nothing else; i.e., every pan is "composite."
The Stainless line pans are composite in the sense that a stainless pan is welded to a diffuser disk. The disk serves two purposes: It spreads the heat evenly on the bottom of the pan reducing "hot spots." It helps prevent warping -- and then helps keep it under control once it starts.
The Tri-Ply line pans are made by All Clad, and are their budget tri-ply line. They are composite in the sense that a stainless interior is wrapped in a layer of aluminum from top to bottom; and in turn the aluminum core is wrapped in an exterior of stainless. The tri-ply construction controls the bottom heat as well as the Stainless. And, at least up to the edge, it's no better. But right at the edges, and for the first fraction of an inch above it, the try-ply is significantly better. Above that... well, it doesn't really matter -- because the contents of the pan transfer more heat to the pan than vice versa. However, if it did matter the tri-ply would be better there as well. The tri-ply is also more warp resistant. Most people find its smoother silhouette better looking -- and fwiw, it's more fashionable.
Neither will cook faster than the other. Neither is meaningfully more efficient than the other in terms of "insulation," or using "more fuel," or anything like that. The Tri-Ply will help you with certain kinds of cooking and control unintentional burning a little.
If you can afford the tri-ply without breaking the bank, it's worth the extra hundred buckaroos. In fact, Emerilware Tri-Ply is one of the real bargains in cookware. Compared with the prestige brands like All-Clad, Calphalon, and Gourmet Standard, you get 92% of the rich, buttery goodness for 48% of the price. This is the one people who can afford a lot look at and wonder whether they should spend it. At least in a sane world, they do. Although it's not what I have, this is the first thing I recommend when people ask about stainless sets and introduce the subject of money in the same sentence.
If affording the tri-ply means giving up something else you really want, don't shed hot tears. The "Stainless" will last more than long enough for you to afford to replace it with one of those crazy choices -- and it too is a good deal. Disk bottom cookware might be a little old-hat, but it still performs well enough.
With due deference to Dillbert, I wouldn't worry about copper cookware. Just like the difference between Stainless and Tri-Ply, it won't make a good cook out of a poor one. It has benefits, but its price, for most people, puts it well into the area where the Law of Diminishing Returns is enforced.
Congratulations! Best wishes! .
Hope this helps,