Are "part of speech" and "syntactic type" the same concept? If not, what are their differences?
The term part of speech refers to the traditional word categories. Usually, somewhere between eight to twelve basic parts of speech are posited, e.g. noun, verb, adjective, adverb, adposition (preposition, postposition, circumposition), coordinate conjunction, subordinate conjunction, interjection, ...
The term syntactic type is simply vague. It could refer to anything that is assumed in the domain of syntax. It could refer to words, phrases, clauses, other specific word combinations. I have no clear associations with what is meant when I see syntactic type out of context.
Parts of speech are lexical categories, such as noun, verb, adjective.
With phrase structure grammars, syntactic types, or syntactic categories, are categories of phrases. of well formed sentence fragments, such as noun phrase, verb phrase, prepositional phrase. They correspond to what is called non-terminal in formal grammars, notably context-free.
Parts of speech are thus also syntactic types. It is not the same concept, but there is a containment relation between the two concepts.
However, in lexicalized grammars, phrasal constituents are always associated with a lexical categories, so that syntactic types no longer have much role, or at best have a lesser role.
Syntactic concepts are somewhat dependent on the syntactic theory and formalisation used. This is obviously to be expected.
The differences are not as clearcut as may seem
By lexical, I do mean related to word. However, a word can be a phrase, so that if lexical applies to words, it can apply to some phrases, precisely those that are composed of a word.
Actually, it may be a little bit more complex, if we want to keep things consistent between synthetic and isolating languages. Then I am not sure that lexical should pertain to single word in the context of an isolating language (though that seems to be the case from what I read, or do not read).
Or, taking an opposite viewpoint, if lexical elements (lexemes) are to be composed of a single word (but, what is a word?), then what is considered lexical category in a synthetic language may be considered only as a syntactic category in a more isolating language.(see also: Is there really a difference between agglutinative and non-agglutinative languages when spoken?)
Following this discussion, the issue of what is lexical, or maybe of what is part of speech independently of lexicality, is not completely clear to me.
Indeed the issue is not completely clear at all according to Haspelmath in "The indeterminacy of word segmentation and the nature of morphology and syntax" (Folia Linguistica). Rather than looking for a difference between wordhood and lexicality, he questions the very concept of word to "conclude that we do not currently have a good basis for dividing the domain of morphosyntax into "morphology" and "syntax", and that linguists should be very careful with general claims that make crucial reference to a cross-linguistic "word" notion."
The issue of defining lexical categories raises other problems, such as addressed by Grimes in "Refining Parts-of-speech in the Lexicon", that lead him to state that "parts-of-speech for some parts of the lexicon may need to be defined syntactically, rather than lexically. After all, the whole notion of parts-of-speech is with reference to the syntax of a language."
Doing so might escape the consistency problem between isolating and synthetic languages.
I am not sure where the consensus lies regarding the possible variations in these definitions. There may be better characterizations, or other definitions. Being consistent across the variety of languages seems a daunting task.