The week after this one is 'next week'. You don't use the article when you are speaking in 'real time'. I'm on holiday next week. I'll phone you next week. We use the article ('the next week') to refer to the week following a week other than the current one - or in other words, not 'real time'. For example: 'The history course lasts three weeks. The first week focuses on political history and the next week focuses on social history..'

Yes, there is a difference. Usually, we say "next week" or "next year" without the article -- meaning: the week after this one -- eg: "see you next week." We only use the article to refer to the week after a specific week other than this week. Example: "Peter called him during the first week of July, and again the next week." "Sam told Mary (on Christmas) that he'd be arriving the next week."