There are subtle differences. Your examples illustrate the differences.
Picking strawberries makes me think of pick-your-own farms, a day-out for the family. It is not a professional activity.
Fruit pickers are people who pick fruit for a living. They tend to move from crop to crop according to which fruit is in season. When strawberries are in season, they could be called strawberry pickers and strawberry picking is indeed a strenuous job.
The picking of strawberries puts the focus on the picking which here means selection, not harvesting. Picking strawberries for a top restaurant requires skill and patience. It is a different kind of job.