"Continuous" means uninterrupted, whether of time or space as in "continuous rain" or "continuous range of the mountains". For example: "A continuous white line in the middle of the road means no overtaking". "Consecutive" describes events, numbers, etc. that follow one after another without an interruption. For example: "This is the fifth consecutive weekend that I've spent working, and I'm a bit fed up with it". So if you are talking about something discrete you should use "consecutive" like days, but if event or object or something else can't be divided, use "continuous". So usually when people are talking about days, they use "consecutive days".

As for me, consecutive means more like something happened and ended and then it happened again and again. For example, "Lakers had 3 consecutive wins". They win and win and win. "Sam ran for 3 hours continuously". He never stopped running in that three hours.