Data can be open but not linked, or linked but not open.
"Linked data" refers to data that is machine readable, semantic data, that a machine can 'understand'. The "semantic meaning" comes from the links, hence the names. "Open Data" refers to data that is accessible to anyone (e.g. without monetary cost to access) with a permissive license on reuse (e.g. public domain or CC0).
Development in linked data usually focuses on tools that provide meaning to data as microdata, RDFa, or RDF, and ontologies that provide meaning of terms. Open data focuses instead on tools that allow users to access the data conveniently, focusing on tools such as RESTful APIs, and formats that allow a user to query and subset data such as JSON or XML.
Tim Berners-Lee suggested this five star rating system to help think about linked data:
★ make your stuff available on the Web (whatever format) under an open license
★★ make it available as structured data (e.g., Excel instead of image scan of a table)
★★★ use non-proprietary formats (e.g., CSV instead of Excel)
★★★★ use URIs to denote things, so that people can point at your stuff
★★★★★ link your data to other data to provide context
In Tim's system, five-star linked data has to be open (first star, also open format, third star); The linking is really in stars four and five.
By contrast, most open data systems provide the first three stars (e.g. a RESTful API providing data in JSON format), but don't necessarily hit the fourth and fifth star. From this we might surmise that all linked data is open data, while all open data is not linked data.
One could arguably claim that they provide linked data (URIs, links to other data) behind some proprietary firewall for internal use only, and thus it is not open data.
