Why is a heated bed important? I hear that heated beds can help with removing finished prints, but not all printers have them! 


Is this a nice to have or must have feature? 
Are there any downsides to heated beds?
I'll try to give it a shot as the other answer (perfectly sound answer b.t.w.) does explain "how" we use heated beds, but not "why" they are actually needed for good prints.

Plastics or polymers are mostly amorphous (no macro crystalline structure) and usually relatively hard and brittle at low temperatures (this is referred to as "glassy"). By increasing the temperature of the plastics the state of the material changes as it becomes soft and more ductile (called the viscous or rubbery state). The temperature at which this transition takes place is called the "glass transition temperature". It is this temperature where you would heat your bed too (or close to this temperature). Below the glass temperature, the expansion is reduced/low because the polymer chains cannot move easily due to the closer packing and stronger inter-molecular forces; increasing the temperature increases the expansion lowering inter-molecular forces. The difference in stress between the bed and your print is therefore reduced (as of the expansion  of the heated polymer).

This explains why we use heated bed to get the prints to stick to the bed. To answer your question why it helps in removing prints is that when the temperature is lowered, the polymer gets back into its "glassy" state, shrinks a bit and automatically loosens the print from the bed. 

Indeed for printing PLA you can print without a heated bed, but for materials with a higher glass transition temperature and higher printing temperature it's almost a must to have. Downsides of heated beds are that they require a lot of power and when improperly installed lead to burned wires and connectors. It's usually wise to strain relief the wires from the bed and use proper gauge silicone leads and connectors.