Both words mean the same thing (a sickness), but 'ailment' (by definition) is more likely used to describe a minor illness e.g. a stomach ache, where as 'illness' is a broader term and can be used for any type of sickness and possibly more long term sicknesses e.g. flu, cancer

To add to the answer below, "ailment" is much more old fashioned and/or literary (something you'd find in old novels or grandiose speeches). Modern professional/purely practical English would pretty much always use "illness".