I am writing the answer according to my current understanding
Distributional semantics are computed from context statistics.
It is clear from the statement that the embedding of a word, in case of distributional semantics, is computed from the context statistics i.e., based on the contexts in which the word occurs.
Distributed semantics are a related but distinct idea: that meaning
can be represented by numerical vectors rather than symbolic
This means that embeddings obtained from distributed semantics mayn't be obtained from the symbolic structures. Now, we need to understand what is meant by symbolic structure in this case. It can be same as that of context of a word. We can understand it from the following definition of symbol structure
A physical symbol system consists of a set of entities, called
symbols, which are physical patterns that can occur as components of
another type of entity called an expression (symbol structure). Thus a
symbol structure is composed of a number of instances (or tokens) of
symbols related in some physical way (such as one token being next to
So, it can be understood that distributed semantics are the embeddings obtained not only through the context statistics as in the case of distributional semantics. For example, there are distributed representations beyond distributional statistics, in which embeddings are calculated from the internal structure of words and not from the context in which the word occurs (p 341). One can understand it from the following excerpt from the same page
How can word-internal structure be incorporated into word
representations? One approach is to construct word representations from embeddings of the characters or morphemes.
Thus, to be concise, the embedding for the word
cat is only obtained using the context statistics of
cat in case of distributional semantics and in case of distributed semantics, the embedding of the word
millicuries can be calculated from the embeddings the morphemes $milli,curie,s$ rather than the context statistics of the word
millicuries since it is a rare word which is unlikely to have reliable context information available.