The Latin adverb 'furtim', which is presumably what is translated as 'furtively' in this connection, means literally 'in the manner of a thief', coming as it does from the noun 'fur', a thief. It thus came to mean 'stealthily', used in connection with actions that someone is trying to keep hidden, behind the backs of others. I have not been able to check precisely where it occurs in Book 4 of the Aeneid, but it might be worth making the general point that Book 4 contains much more of a sense of intrigue than any other of the 11 books of the poem. The love affair between Dido and Aeneas, while not illicit in an objective sense, has the feel of an illicit liaison on account of the guilt that both partners feel because their greater destiny is the foundation of great cities and the establishment of their people on a secure homeland. Their spontaneous passion for each other, consummated in a chance thunderstorm and before their respective followers have had a chance to anticipate any formal union between them, creates a sense of unease and disquiet from the start and this grows as the story progresses. Messages through the intermediary Anna, the malign influence of rumour and 'fama', Dido's growing anticipation of betrayal...all these help to create an atmosphere in which the word 'furtim' seems entirely at home.
Furtivemeans stealthy or surreptitious or clandestine. If a person does something furtively, the suggestion is that he probably has hidden motives or that he acts in a secretive sort of way. A person who steals money from the place where he works would do so furtively, so that no one would know what he is doing. Prisoners would plan their escape from jail furtively so that the guards won't be suspicious about what they are up to. To explain Virgil's use of the word, I would need to know the passage in which you find it.
"Furtive" means secretive. Aeneas uses the word when he is talking to Dido. (Virgil is the author of The Aeneid and also a character. In Book IV, it is Aeneas who is using the word.)
Aeneas says to Dido, who has accused him of trying to run away without telling her what he was up to. Aeneas responds, "I never hoped to hide - do not imagine that - my flight; I am not furtive" (4.455-7). However, Aeneas is being "furtive" and his usual self-assured manner falters when he comes under Dido's direct and emotional fire.