It appears no one here has actually read the essay. He's not talking about mathematical complexity of Maxwell's equations (and no, you can use differential forms, Clifford algebras, quaternions, vector calculus, but no matter how you write it, it's still the same thing, and at the end of the day, you will end up solving exactly the same set of coupled partial differential equations, to the letter --there's no magical mathematical notation that makes this go away).
The difficulty he's referring to is in the physics (and not mathematics) associated with the idea that fields are real, fundamental physical entities, and cannot be reduced to mechanical models with "gears and wheels" permeating the space (which was a popular idea back then).
You prompted me to read the actual essay. It was considerably more interesting than the comments here would have suggested; which is not a surprise, given it was penned by Freeman Dyson.
His insight that Maxwell's contemporaries lacked even the language to fully describe what a transformative idea he had is acutely interesting. Once again language both shapes and traps thinking.
He makes the same point with quantum mechanics as well - that we're constrained by not having the proper language to describe the most fundamental behaviour. I think it's fair to say that point still stands today.
The difficulty he's referring to is in the physics (and not mathematics) associated with the idea that fields are real, fundamental physical entities, and cannot be reduced to mechanical models with "gears and wheels" permeating the space (which was a popular idea back then).