Taste
This is the strongest argument, and it is mechanical. Fruit destined for transport is picked before ripeness to survive the journey. But many fruits — strawberry, peach, melon, tomato — produce no further sugar after picking. They soften and change colour, but gain no flavour.
Seasonal, local fruit can be picked ripe. That is the whole difference.
Nutrition
Some vitamins, vitamin C in particular, decline after harvest. A seasonal vegetable bought soon after picking retains more than a transported and stored one.
A useful nuance: frozen produce is frozen within hours of harvest and retains those vitamins very well. Frozen vegetables often beat "fresh" ones transported and left three days in a fridge.
On carbon footprint, the usual reasoning is wrong. Transport is a small share of a food's impact; production method weighs far more. A local tomato from a heated greenhouse in winter has a higher footprint than an imported field-grown one. "Local" and "in season" are not synonyms.
Constraint as method
Seasonality's real contribution may lie elsewhere: it imposes a constraint, and constraint makes people cook. An imposed box forces you to find something to do with what you would not have chosen — which is where most regional cuisines came from.
The real calendar
It is shorter than people think. Asparagus lasts six weeks, strawberries two months, cherries three weeks. For much of the year in northern Europe, the season means roots, squash, brassicas and preserves.
our piece on food design sets out how the form and colour of food alter perceived taste.