The central difference between linguistic determinism and linguistic relativity is the idea that world-view concepts and thoughts cannot be altered versus can be altered.
Edward Sapir, a linguist, studied indigenous languages. These are languages of indigneous people who are the first peoples occupying a given land. For instance, Native Americans, Inuit, and Aboriginals are three groups of indigenous peoples from the North American continent, extreme northern climates, and Australia, respectively.
In his studies, Sapir was surprised at the contrasts between how indigenous people and European people spoke about the order of the world. Examples are time as something that flows versus something that is quantifiable and linear, and relationships between people in reference to hierarchy and social proximity. This discovery led him to the conclusion that the language we are born into determines in a fixed and unalterable way the way in which we perceive objects, manipulate object, understand abstractions and our relationships to abstractions. Here's an example about feasting from his work. The English thought is six separate words focused on the person (HE), while the Nootka thought is one word (with five suffixes) focused on the action (BOIL):
English, six words: HE invites people to a feast.
Nootka, one word with five suffixes: Tl'imsh - ya - 'is - ita - ma. [Translated to English: BOIL - ed - eat - ers - go-for - he-does.]
Sapir's student, Benjamin Whorf, dedicated himself to proving and examining Sapir's hypothesis. In his studies, he discovered that while there are deterministic constructions of reality imposed on our cognition and perceptions through our birth-language, there are also cultural factors (cultural-linguistic factors) that can override aspects of the deterministic strictures of our original language (or languages, if bi- or multi-lingual), and thus can enlarge our perceptions and cognition relevant to the world we experience.
According to Whorf, since there is an opportunity to broaden and enlarge the deterministic structure of language (at least in degrees), then the deterministic nature of language is relativistic: it is relative to what is done or what is experienced to broaden or enlarge perception and cognition.
As an example, the Japanese have an expression that is given as a sort of blessing when someone leaves home to go anywhere, to work, to school, to the market, anywhere. The person departing says what is loosely translated into English as "I go come back." This expresses a whole cycle of a journey: a journey starts at the door and end when the door is returned to.
The person staying at home says, loosely in English, "Off you go." It is a blessing over the person that hurries them on their journey--it doesn't impede them--with the happy anticipation of the culmination in the return.
In English, there is no continuity between going and coming back: we always hope there will be a coming back. We don't assume a coming back; we can only hope (sometimes in anguish if a parent) for a coming back. When an English speaker learns this cultural-linguistic element of Japanese world-view, it comprises different cognition and can be liberating.
The great difference, then, is that Sapir's contribution of linguistic determinism sees only the rigidity of the contrasts between languages, while Whorf's contribution of linguistic relativism acknowledges that conceptual paradigms of thought can be modified, expanded, enlarged and reshaped relative to the experience the individual has or the effort the individual makes.