Headers or axis
Discrete fields take distinct, separate values. Tableau draws them as headers — one pane or label per value. Continuous fields can take any value in a range, so Tableau draws an axis.
That single difference explains most of the surprising behaviour beginners run into. A discrete measure on Columns gives you one header per distinct number; the same measure as continuous gives you a proper numeric axis.
Dates make this concrete
Dates are where it matters most, because Tableau offers both forms in the same dropdown.
Discrete date parts — YEAR(Order Date), MONTH(Order Date) — pull out a component and ignore the rest. MONTH as a discrete part gives you twelve headers, January through December, with every year pooled together. That's what you want for seasonality.
Continuous date values — MONTH(Order Date) as a truncated value — keep the position on the timeline. May 2023 and May 2024 stay separate. That's what you want for trend.
Practical consequences
Discrete fields can be sorted manually and reordered; continuous fields follow their natural numeric order. Discrete fields get a colour legend with distinct swatches; continuous fields get a gradient.
Filters differ too: a discrete filter shows a checkbox list, a continuous filter shows a range slider.