Differentially/Differently

[upinaballoon]

Recently I have encountered 'differentially' written or spoken where I would use 'differently'.
Before I go the dictionary, which is not modern, I feel that the word 'differential' is a word that would be used by 'blokes who know about engines and all that'.
Are they two different words or are they alternatives?

It’s mostly a technical term, but also to do with ratios of things apparently so definitely different from differently. They’re not alternatives. It sounds a bit like people using words to try to show off.

differentially
adverb
dif·fer·en·tial·ly ¦difə¦renchəlē -li
: in a differential manner : so as to constitute or create a differential : so as to make distinction between different individuals or types

Can I add 'purposely' and 'purposefully'?

Purposely vs. Purposefully: Which is Right? | Merriam-Webster

They are indeed two different words (not differential words). I see 'differential' as addressing the distinction between things in the same category, while 'difference' is the distinction between things in different categories. This can be subtle, depending on context. In your engineering example, differential refers to things like the way cogs can move at different speeds though driven by the same shaft. While cogs and, say, pistons are different things. Medical uses include differential diagnosis - a list of each possible diagnosis but they're all diagnoses, and differential cell counts - amounts of types of blood cells in a sample but they're all blood cells. While diagnosis and sampling are different procedures. I wasn't even sure that 'differentially' isn't a made up word, but the dictionary says it exists.

As far as I'm concerned 'purposely/purposefully' can definitely be added and thank you for the reference and I definitely think of them as having slightly different meanings. I do hear the use of 'purposefully' and think, "No, you need 'purposely. That is enough."