Where it breaks is fairly important.
First and foremost, if the tether breaks, everything above the break-point will "fall" upwards, escaping into space. Since most of the dangerous environments are near the bottom of the tether, only a short bit will collapse back down to Earth.
A space elevator needs to be fairly light by necessity.
The nominal (20 ton) Space Elevator tether weighs about 10 grams (a third of an ounce) per meter, and so over any square kilometer, the amount of material will be miniscule.
You definitely don't want to be climbing it when it happens though.
Finally, there will 4-5 climbers on the tether. While the top ones may be able to remain in orbit, the rest will start falling towards the Earth. Manned climbers will of course have the ability to soft-land, and cargo climber might be allowed to simply splash down.
Above the break the counterweight would "fall" away from earth into a higher orbit.
It probably wouldn't be terribly spectacular with a break near the ground.
If you want something which is likely to fail in a really spectacular and devastating fashion (while also having an advantage over a space elevator that we could actually build one using known materials ) then you might want to look into a space fountain:
I decided to try to work out the energy involved if there was a break very far up and most of the elevator fell to earth:
In theory a space elevator might weight anything between 20 and 750 tons.
I can't do the math for the real case where the mass is spread out all the way up it's length but I'll err on the side of "more disastrous" and do the math for all the mass being higher up near geo.
Velocity at Geo = 3.07 km/s.
(kinetic energy of 750 tons at 3.07 km/s) = 3.206×10^12 J (joules)
which is about 95% of the maximum fuel energy of an Airbus A330-300 (97,530 liters of Jet A-1) or 1/20 of a little boy bomb.
The potential energy of 750 tons falling from 35000km is larger but still only in the range of .05 megatons and most of that would be lost to friction with the atmosphere.
Imagine spreading .05 megatons of TNT along a line 20,000 km long (still erring on the side of "more disaster"), that gives you 2.5 tons of TNT per km or put another way, 2.5 kg of TNT per meter.
Imagine a stick of dynamite about 5.6 cm's thick that's 20,000 km long. That's the total energy of the falling space elevator.
You wouldn't actually want that hitting you but if you were 50 yards away from the line you'd probably be fine. Again, that's even pretending no loss of energy to the atmosphere as it falls.
A falling space elevator wouldn't be great but more along the lines of "it's bad to be hit by things falling on you" rather than "everyone nearby is very dead"