So, a pickup creates a magnetic pull on the strings, and this does have an effect on the way the strings are vibrating. It's an experiment you can do yourself: You can adjust the height of the pickup so that it gets closer to the strings. If it's close enough, you will hear a vibrato-effect when you're playing. Well, that's undesirable, unless you want to have vibrato everywhere.
So, if there's more magnetic pull on the strings, there's a higher possibility for unwanted vibrato; this could mean that you need to adjust the height, so that the pickups are lower (further away from the strings). Lower means a weaker signal/output; and the different strings might be affected differently, so this means different tone overall.