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Weak pullups are often fine for digital inputs, such as a switch a user might push. A strong pull-up may be needed on something like an I2C bus. If there is a fair amount of capacitance on the line, the pullups may take a long time to pull the line back to a logic HIGH which would limit the maximum bus speed. By increasing the strength of the pullup, the bus is pulled up quicker, and the speed can be increased.
As a very fat thumbed rule of thumb, I start with a 10k pullup and work from there. In battery powered applications I'll try to go up to 100k or more if I can get away with it, and in bus communications I've gone to a couple hundred ohms. Cell phones use pullups in the mega-ohm range.