They sound exactly the same, but you're right that it can't be julen here because of grammar. This is where gender is somewhat helpful. It's ett hjul but en jul. So julen is single definite, which means it would have to be julen är vit 'the Christmas is white'.
en jul, julen; jular, jularna 'a Christmas, the Christmas; Christmases, the Christmases'
ett hjul, hjulet; hjul, hjulen 'a wheel, the wheel; wheels, the wheels'
When I just started learning Swedish, having to conjugate the adjectives was a liability, but now it's an asset! Not knowing the proper declension of "hjul", I was going to write: "the wheel is white", but then I noticed the plural suffix on the adjective, and realised "hjulen" is plural!