Basically one is correct and one isn't. The accepted term in both psychology and literary usage is "internal monologue." A dialogue is between two people. A monologue involves one person.
Although the term internal dialogue is sometimes used by people, it isn't the correct usage, unless of course your characters have multiple personalities.
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Internal monologue is the inner voice that narrates one's thoughts and feelings. Internal dialogue involves a conversation within oneself between multiple perspectives or voices.