- Copper is toxic. You have to get the inside lined with tin. Tin has a very low melting point and can melt during cooking. Even if it doesn't, it wears off with usage and the pan has to be lined again. I don't know how easy it is to find somebody who lines, but nowadays, it is not so common. Alternatively, you can buy a copper pan with a thin internal stainless steel shell, which doesn't need re-lining. You have to buy it from a good brand to be sure that what you get is mostly copper (as opposed to mostly stainless steel) and you lose the advantage of copper's great heat diffusivity.
- Copper is a reactive metal, you can't put it in the dishwasher (although, it shouldn't need more maintenance than cast iron - it is just more problematic than stainless steel).
- Copper is an expensive metal. Due to the low demand for copper pans, they are a niche product for people who want them very much, and this increases their price even more. Expect to pay €200 to 300 for a copper pan.
- you can't use a copper pan on induction. You can use it on gas, but it will likely discolor/blacken.
- copper is heavy. It has even higher density than cast iron (although the difference isn't that big).
- Copper has some great heating qualities. In fact, it is the best metal for cooking (maybe silver is better, but I haven't heard of silver pans and don't have the numbers for it). It has five times the thermal conductivity of iron. The problem is, after you have taken all the relevant variables into account, you arrive at about the same specific heat per cubic centimeter as iron. So, when heated at a constant temperature, a pan of copper will give you the same heating evenness as an iron pan of the same thickness. If the heat changes, the copper pan will react much quicker, which is a good thing. The catch: I don't think anybody makes copper pans the same thickness as iron pans, and a thin copper pan will heat nowhere as well as a thick iron pan.
Example: look at this pan. This is a €220 copper pan, 2 mm thick. (Don't ever buy "copper" pans which don't tell you the thickness, they are mostly stainless stell with a thin copper coating on the outside). My €15 cast iron pan is 5 mm thick. It doesn't have the quick response of copper, but it should beat the thin copper pan in slow, even heating and heat retention.
The bottom line: I haven't tried a copper pan, but wouldn't do it because of the price, even though they might heat better than iron. If you want to invest the money, you are welcome to try them and tell us how it went. I have never heard of bronze pans outside of historical contexts, but I guess they will share the downsides of copper (except maybe being cheaper), and I would have to see the numbers for the heating qualities.