Conjunctions

Синцов заворочался и проснулся с той же мыслью, с какой заснул.
My humble knowledge of Russian grammar tells me that a declension of "который" should be used, instead of a declension of "какой"; that is to say, it should be "... с которой заснул" instead of "...с какой заснул".
Also I often come across constructions like: "Я знаю того, с кем вы говорили" [instead of "... с которым"]. This one seems to be more common in spoken Russian.
My first guess is that the meanings are the same, and the difference is in the degree of formality. What are the grammatical nuances here?

Который

Какой

Interrogative pronouns (вопросительное местоимение) который and какой

который -- "which" (in a sequence); or "which one exactly"
какой -- "what kind of"

Relative pronouns (относительное местоимение) который and какой

который -- "who, which"
какой -- unlike который, which relates to specific objects and persons, какой relates to things and persons of a particular type

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кто functions as relative pronoun to тот (тот, кто "he/the one who"), те (те, кто "those who"), никто (никто, кто "no one who"), все (все, кто "everyone who"), первый (первый, кто пришёл "the first to come"), единственный (Он же единственный, кто на нас постоянно жалуется "he is the only one who constantly complains about us")

In the first phrase какой related with той and translated together as "the same as"; the word который gives less relations between clauses.
Imho the word который has identical meaning with the word "which", If translate your second phrase as - "I know the man to whom you talked - it's obvious that you cannot use "which".

Quote:

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Originally Posted by Misha Tal

Синцов заворочался и проснулся с той же мыслью, с какой заснул.
My humble knowledge of Russian grammar tells me that a declension of "который" should be used, instead of a declension of "какой"; that is to say, it should be "... с которой заснул" instead of "...с какой заснул".

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This is an archaic phrase. In contemporary Russian "той, которой" is better.

Quote:

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Originally Posted by Misha Tal

"Я знаю того, с кем вы говорили"

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As was told, "тот, кто" is applicable in the sense of "the one who".

Another archaic phrase is "с той, что", "с тем, что" can be used in both senses in both sentences above.