They are more or less synonyms to most people, but theft is a bit broader. Larceny has fallen out of favor because it's an older term and its definition required that the property be physical and of a nature that it could be taken away. So "theft" allows services, land, etc.... to be included.
Robbery is theft when there is direct personal contact made with the victim, usually with violence or threat of violence.
Burglary is the entering an area with the intention of theft.
In most states the line between petit and grand larceny/theft is $500, but that sometimes changes.