What is the difference between a conjunction and a linking verb?

Conjunction is a part of speech used to connect words, phrases, or clauses. A linking verb is a verb (usually a BE-verb) used to connect the subject to its predicate nominative/adjective.

Conjunction used to connect words: Michael and Stephen are joining the contest.And connects "Michael" and Stephen".

Conjunction used to connect phrases: The children were running around and making so much noise.And connects "running around" and "making so much noise".

Conjunction used to connect clauses:Either Marie will take summer classes or her father will send her to the province.Either ... or connects "Marie will take summer classes" and "her father will send her to the province".

Linking verb joining subject to predicate nominative: Mr. Castro is the new president of the company.Is connects "Mr. Castro" and "president".

Linking verb joining subject to predicate adjective: Carol is very talented.Is connects "Carol" and "talented".

Mom's cake smells delicious.Smells connects "cake" and "delicious".

Note: Not all linking are BE-verbs as in the case of the word "smells".