I'm wondering what causes the price difference between a Serial-to-RJ45 adapter and a Serial-to-Ethernet server. Given a serial-interfaced instrument, can one use either to connect the instrument to a computer? (Of course, the first must be wire correctly first).
To answer your first question: the Serial-to-RJ45 adapter is just wires, the Serial-to-Ethernet server is a computer in a box.
I can't answer your second question without knowing more about your equipment, but perhaps this information will help you.
This adapter to me looks like it is designed to permit users to use inexpensive ethernet cables in place of relatively rare serial cables. In order for it to work, you need to have a serial port on your computer. The ethernet jack on your PC will not output serial signals and this adapter will not turn them into serial signals.
This is a small computer which connects to your local network and has a serial port which you can connect equipment to. It comes with software that allows you to use it as if your computer had the serial port itself.
If your computer doesn't have a serial port, USB-to-serial adapters are available very cheaply
There are a lot of different serial communication protocols. Generally "serial" on an instrument means RS-232 or RS-485, but could be something different. Make sure you understand the requirements of your equipment before buying something that may not work.