Will upgrading my USB-to-ethernet adapter to USB 3.0 make any difference?

I think this depends on your internet connection. You can check your download (and upload) speeds at speedtest.net.

Since USB 2.0 has a maximum speed of 480Mbit/s, your internet speed will have to be faster than this to make it useful to get a USB 3.0 adapter (which supports up to 4.8Gbit/s).

USB 2.0 only does half-duplex, so you'll get more latency. USB 3.0 can do full duplex. If you are doing something like gaming that is latency sensitive, it's probably worth finding a full-duplex solution (make sure your USB 3.0 device can do full duplex). Bandwidth wise, you'll need a wide pipe to the internet to need USB 3.0 over 2.0 based on USB max throughput.