I agree with replies above. It is VERY likely that the USB ports you have on that old case are USB2. The ones used for USB3 are different so they must be replaced somehow, AND the cable to them from the mobo will need to be changed, too. Of course, your mobo will need to have the right USB 3 header.
FYI, on newer cases it is common that the plastic inserts in the socket carrying four contact strips for USB2 are BLACK. For USB3 sockets they are BLUE, and if you look closely into them you will see five added contact points at the back of the socket.
On naming, the names certainly have changed rapidly with the result that, while there is a fixed current standard naming system, MANY products on the market have older names or mixed-up labels. USB2 is still there, almost always now as USB2.1 The new system now is all USB 3.2, Genx. (I'm not getting into USB4.) Within USB 3.2, the Gen1 system has a max data transfer rate of 5 Gb/s; Gen2 max is 10 Gb/s; Gen2x2 is 20 Gb/s. All USB3.2 ports are supposed to provide power at 5 VDC up to 0.9 A per port, although some have been made for higher max for devices designed for that. (It IS a bit confusing when you get Hubs containing CHARGING ports that look like regular USB 3.2 ports, BUT are really NOT for data transfer - the are only for higher charging currents.) We should note that all USB2 and USB 3.2 Gen1 and Gen2 HEADERS on mobos provide TWO ports her header, so the cables from them connect normally to two exposed sockets. The Gen2x2 mobo headers are different because they essentially combine the capabilities of two ports to provide a SINGLE port able to reach 20 Gb/s. The mobo headers for USB2 , for USB 3.2 Gen1 or 2, and for USB 3.2 Gen2x2 are all different.
The most common exposed connector style is Type A. Although the ones for USB2 and USB3 are different, they do look similar at first glance (but see above). The newer and smaller connector is Type C. The USB people say that a Type A connector can work for all of these because each is designed with backwards compatibility. BUT they say clearly that you can NOT rely in getting the full higher speeds of Gen2 or Gen2x2 using a Type A connector. Only the Type C connector system is expected to deliver that full performance.
So, OP, what do you need to install? You'll need to replace the front sockets and the cables that feed them. With what? depends on what you have on your mobo now, and on what you might want to plan for in future. If your mobo USB3 header is the Gen1 type, then sockets of the type A only is sufficient for now because that's all the performance your mobo can provide. If the headers can do Gen2, OR if you plan that you might get another mobo with that capacity, you should probably try to install both Type A and Type C ports on the front. Type A's for so MANY existing devices to plug in, and Type C for any newer Gen2 devices. IF your mobo has a new Type E header for a Gen2 or Gen2x2 port, you MAY want to add a cable and Type C front port from that for future use. Right now there are few devices that can actually use Gen 2x2 speeds, but they will come.