I designed a logo for a client. He liked the idea it presented, but wanted to see alternative ways to represent its idea, so I came up with several new designs of similar concept. I now have to package the new designs into a presentation document, and when it came to titling its sections, I wondered: are the new designs variants of the initial one, or are they variations?
Coming from a technical perspective, if I report that I tackled a problem using a variant of Algorithm A, then this variant is most likely one that other people have used before and there is literature on its use. If instead I say I was using a variation of Algorithm A, then the existing variants of Algorithm A didn't meet the particular requirements of my problem and I therefore needed to create my own adaptation. People after me might decide to call this a variant of Algorithm A, if they deem it close enough to the original.
Caution: I'm not a native English speaker.