I would suggest planting something more desirable to choke out the weeds. Possibly a creeping sedum or mother of thyme if your yard is hot and dry, or some sort of moss if it's shady and wet.

After 20 years it is probably time to re-grout the bricks.

For getting rid of the existing weeds, a friend of mine uses a blow-torch weeder (I don't know the name of it -- it uses a propane bottle and has a long rod attached that you aim at the weeds). Works very well and non-toxic. Then you could put down polymeric sand (http://www.groundtradesxchange.com/pavers/polymeric_sand.htm) to help keep the weeds from growing back.

I think the torch tool is called Weed Dragon.

The bond between the mortar and brick is broken for some reason and until that is repaired you will continue to have plants growing in that space. There are tubes of latex mortar that could be used to fill in the spaces that do exist, but the best wat to solve this is to pull up and relay the brick, properly.