$\begingroup$
An o-ring usually (but not always) has a round section and is made of a compressible material such as rubber, neoprene, silicon etc.
A washer tends to have a square or rectangular section and is usually made of a harder material brass, copper, aluminium, steel etc.
It is not about advantages or disadvantages, but which is used will depend on the characteristics of the purpose, so if it is a delicate device at low pressure then a softer material can be used especially if the units are designed with matching channels for the sealing component to "sit" in.
Many times a copper washer can be used - some car sump plugs have copper washers for sealing, others just use machined steel faces...
Just to muddy the waters as it were, there are o-rings called Wills rings which are a hollow metal torus filled with an inert gas, these are capable of sealing at high temperature and pressure. One use was as part of an engine head gasket sealing the head to the block around the combustion chamber...
For more information o ring box, please get in touch with us!