A standard is typically specifications / parameters applicable to many things or types of things. An example of a standard would be ISO 9001 which is applicable to just about any organization. Another example would be ISO 10110-17:2004 (Optics and photonics – Preparation of drawings for optical elements and systems – Part 17: Laser irradiation damage threshold) which is applicable to any such devices any company makes (manufactures).
A specification is typically a company specific document which sets parameters for an item. An example of a specification would be geometric, electrical and other parameters for a specific device a specific manufacturer makes. However, a company (or other entity such as a government) may develop a specification for somethng such as a device or part which will be purchased. Thus, a specification may be developed for many companies as a requirement by one company (or a few companies) which is applicable to a device or part procuced by any company accepting a contract.