Discrete fields draw headers; continuous fields draw axes.
When you want to add some reference line in your date field then go with continuous. if you are in discrete then it will not allow to create ref line.
Notice that there is a discrete header for each month.
In the second chart, I am using the same exact data, but I have changed the Date dimension from discrete to continuous:
As you can see, I now have a continuous axis of time. Since the axis is continuous, I cannot change the order of the dates; they follow a chronological order from oldest date on the left to most recent date on the right. On the other hand, when the Date dimension is being used as discrete as pictured in the first image, I am able change the order of the dates. For example, I could sort the bars in descending order, with the month with the highest sales first, and the month with the lowest sales last. Which brings me to my second rule when determining whether I should use a field as discrete or continuous:
Discrete fields can be sorted; continuous fields cannot
So how you might use this in the real world. If you know that you want to look at a trend over a continuous period of time, you would want to use a continuous date, which will be colored green on the view. If your analysis requires you to have discrete marks that can be sorted, you would use the field as discrete, which will be colored blue on the view.