The terms can be used interchangeably. If I am not mistaken, mental health practitioners refers to most (or all) conditions as "disorders", following how they are being listed in American Psychology Association (APA) Diagnostic and Statistical Manual of Mental Health Disorders. Personally, I feel that "mental illness" carries a negative connotation while "mental disorder" is a more neutral term. But this is just me.
From my perspective, a mental disorder is a change in a way a person thinks and feels to an extent that it impedes his ability to carry out everyday activities. This can include, depression, anxiety, substance abuse etc. As for mental illness, it is a term sometimes used to describe conditions, such as schizophrenia which has a strong biological basis but is not at all confirmed.
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Yes people always group the illness and disorder together.