Learn English – working on?/with? his laptop

Either could be used and neither would be wrong. The meanings are similar, but there may be some slight differences.

Normally, I would use on. When I am word processing, or checking email, or working on a spreadsheet, I am usually inclined to say:

I am working on my laptop.

Typically, on is the preposition I'd choose.

However, there are cases where I might change that. Let's say my coworker Linda owns a computer at home, and I know for a fact that her hard disk crashed two days ago. (She's also borrowed a company laptop, but I'm not aware of that yet.) At 10 o'clock, I notice Linda's desk at work is still empty, so I ask Gary about this:

J.R.: "Gary, do you know where Linda is?"
Gary: "Yes, she's working from home today."
J.R.: "Wait – how can that be? She got her computer fixed already?"
Gary: "No. The folks in I.T. loaned her a laptop yesterday."
J.R.: "Ah, I see. She's working with a laptop."

This works, because the phrase working with can be used to mean using. As another example, if I was in a wood shop, I might say:

I'll be working with hand tools today.

if I'm not planning to use any power tools.

This is another one of those cases where prepositions are simply too flexible to say there is always one correct preposition to use.