Subordinating conjunctions (Subjunktionen; presumably what you mean by "subjunctions") and adverbial conjunctions (Konjunktionaladverbien) are types of connectors and as such are functionally similar. Connectors are words that express a semantic relationship between propositions (≈ connects; Konnekte).
Subordinating conjunctions embed one connect (the internal connect) into another one (the external connect). A subordinating conjunction must take the position directly before the internal connect; it is not syntactically integrable into the connect. The internal connect is a verb-last clause.
Adverbial conjunctions (a type of adverbial connectors) impose a paratactic syntax (not one of subordination) on the text. They are syntactically integrable into the connect. In terms of topological field structure, adverbial conjunctions can be positioned in the pre-field and the middle field. Adverbial conjunctions do not affect the position of the verb in either connect, and so generally you will find verb-second clauses. As a result of their integrated status, adverbial conjunctions can occur alongside subordinating (and non-subordinating) conjunctions.
A given lexem may be both a subordinating connector and an adverbial conjunction, but usage varies and which one is needed depends on what you wish to express. In general, subordinating connectors and adverbial conjunctions are clearly distinct and one should not have trouble telling them apart in context.
Damit can occur as a (final) subordinating conjunction: Ich gebe dir Geld, damit du dir ein Eis kaufen kannst (= ... in order for you to be able to buy ice cream). It has a different meaning when used as a (consecutive) adverbial conjunction: Er kauft gerade ein Auto. Damit ist mir klar, dass er reich ist (= With that, it is clear to me that he is rich.) Note the syntactic flexibility: Er kauft gerade ein Auto. Mir ist damit klar, dass er reich ist. (middle field) Er kauft gerade ein Auto und damit ist mir klar, dass er reich ist. (use with a conjunction)
Solange as a subordinating conjunction: Sie war unglücklich, solange sie noch kein Auto hatte (= ... as long as she did not yet have a car). As an adverbial conjunction: Sie hatte ein Jahr lang kein Auto. Solange war sie unglücklich (= During that time, she was unhappy). Note the different structure of these two statements: With subordinating solange, the internal connect is the "background" event to the event in the matrix clause; these roles are reversed with adverbial solange.
Note that adverbial conjunctions (Konjunktionaladverbien) are different from conjunctions (Konjunktionen). Conjunctions, like subordinating conjunctions, are not syntactically integrable into the connect.