Abandon does not mean the same thing. In the example you provided, "refuse" means that someone will not use the computer or cell phone, and implies that someone else is asking them to, or suggesting that they do.
Person 1: You should use a cell phone. They are so convenient.
Person 2: I don't like technology. I refuse to use a cell phone. My home phone is all I need.
If you want to use a synonym for "refuse", you can use "decline", which is a less forceful word. It still means close to the same thing, but "refuse" is a bit more aggressive and can sometimes be accompanied by hostility. "I hate all this new technology! It's ruining our society. I refuse to use a cell phone or computer, and I think we should get rid of them all!" Decline can also be a one-time choice. Someone who refuses to use a cell phone probably never wants to use it. But if someone is declining to use it, they may still want to use it in the future. Decline is also most often used to describe someone's action. "She declined to use my laptop when I offered it." It would be unnatural to say, "I decline the use of your laptop." However, you can use "refuse" in either way. "I refuse to use a cell phone." And also, "He refused to use the computer for his lessons."
I hope this helps. :)