A few extra + Pick points to add to leftaroundabout's answer:
Easier to evenly accent 8ths / 16ths patterns, even when playing fast
Even if you can go as fast with three fingers as you can with a pick, you can get more appropriate accenting from the natural up/down motion of picking when playing 8ths / 16ths than you can when using three fingers. This is because even if you can control your three finger motion to accent every second or fourth pluck, they will be plucking the string in slightly different places, leading to the "threes" feeling leaking through. I've not seen any player who can overcome this.
Playing with two fingers gets around this, but again I am not aware of any player who can play with two fingers at the same speed as you can with a pick and keep it up for the length of a gig if they have many fast songs to play.
Notes can ring for longer before they're re-picked than when they're re-plucked
if you're repeatedly plucking a string with your finger, you will be muting the string for the time it's in contact with your finger. This means for fast patterns on one or two strings, the 'duty cycle' period during which the notes are ringing freely can be quite short, leading to a lack of definition in the line. With a hard pick, the period in contact with the string is shorter, so your notes can ring for longer.
Easier to play a string without muting the one below it
When you pluck with fingers, your finger naturally follows through on to the string below and mutes it (this can be avoided, but it may take a little bit of effort). That's often exactly what you want, but if you want to let a string ring and play some higher notes against it, that's easier to do with a pick.
Gives you a defined sound even if you don't have right hand finger callouses
I just don't play hard enough to build much of a callous on my right hand, so if I want to get a defined clicky sound I need to use a pick. I do also love the sound of the strings clattering against the frets, but that's not always the sound I want - sometimes I want a nice controlled sound, just with a bit more edge.
No need to figure out fingering options
When playing without a pick I use three fingers and a thumb - sometimes that can lead to quite a lot of thinking about the best finger pattern to use. Having those options is great, but sometimes when using a pick there's only one way to do it, which can be a blessed relief!
Easier to sing and pick (slightly personal?)
I am fine strumming away on the guitar and singing; playing a bassline and singing messes with my head if I have to think about right hand finger patterns. If I use a pick it seems to take up less headspace, and singing seems easier.
Having said all this, I'm a fingers player 90% of the time. But to be a well-rounded player there's no doubt in my mind that you need to be aware of what a pick can do for you!