What is the difference between silent and synonymous mutations?
I would say that silent mutations, as they have been defined, are a subset (or perhaps even same as) of synonymous mutations. However, the term, "silent mutation", is not used so much these days. Almost every research paper that I have read pertaining to this topic has always used the term "synonymous mutation" irrespective of whether or not it has a phenotypic effect. It seems that the term, "silent mutation", is confined only to textbooks.

If you consider silent mutations are mutations that are not reflected in the phenotype then there some points that you should consider. 


As canadianer mentioned, these mutations can be present in non-coding region. However, mutations in non-coding regions can also affect the phenotype. Moroever, as they mention, synonymous mutations can cause changes in translation rates because of the codon bias which in turn can lead to phenotypic differences (Kimchi-Sarfaty et al, 2007; Waldman et al, 2011; Buhr et al, 2016). These phenotypic differences can arise due to altered protein concentration or altered folding states. So a synonymous mutation is not essentially "silent"
A non-synonymous mutation may not affect the phenotype (neutral mutation). 


I would say that this definition should not be used unless the majority of the scientific community agrees on redefining the term. Till then, you can possibly use the term, "neutral mutations", to mean mutations that do not affect the phenotype. Remember that neutrality of "neutral mutations" is defined with respect to fitness. However, if your measure of fitness pertains to the molecular function of the gene then this definition works.