In general, あたし sounds casual or childish as compared to わたし. Unlike わたし, you should not use あたし in business settings. This song is an "image song" of a character called Falsita, and according to videos on the net, her normal first-person pronoun is 私(わたし).
アタシヲアイシテ is written in katakana, which implies this particular line belongs to a different context. Have you ever seen an all-katakana sentence like this said by a yandere character or someone in a fury? In this case, アタシヲアイシテ represents her bare andhidden but strong feelings, and that's why she (Falsita) temporarily switched from あたし私 to アタシ.
Similarly, a male character who normally uses 僕 may temporarily switch to 俺 when he is excited or angry. Changing thea first-person pronoun is one of the common ways to show one's emotion.