One common scheme electronic voting machines help prevent is forced votes. A bad guy gets their hand on a single empty ballot and writes the name of the candidate he wants to win on it. He then comes to you and threatens you and your family. Says hand in this pre-filled ballot and bring me back your empty ballot, or else... You comply, he fills out the empty ballot again, and repeats.
The electronic voting machines protect your identity. They allow you to vote anonymously. They provide data integrity that is harder to spoof than paper voting methods. I explicitly asked why they don't just vote on paper ballots like they do in Canada (or the UK as you describe). His response was that we take for granted the inherit trust our societies have to allow us to vote in such a fashion without it being tampered.