What they share
Both end in the soft -ia ending that has dominated girls' names for two decades. Both have enduring nicknames (Liv and Livvy for Olivia; Amy, Mia, and Millie for Amelia). Both pair brilliantly with shorter middle names.
Pick Olivia if…
Pick Olivia if you want the stronger vowel sound and the slightly more cosmopolitan feel. Olivia means 'olive tree' and carries both Shakespearean and Mediterranean associations. Liv as a short form is sharp and modern.
Pick Amelia if…
Pick Amelia if you want the softer, more lyrical version. Amelia means 'work' (from the Germanic 'amal'), but in practice carries strong connotations of independence, partly via Amelia Earhart. Millie as a short form is irresistibly warm.
Olivia skews slightly more polished; Amelia slightly more playful. If you are drawn to the nicknames, Liv suits a more decisive personality, Millie suits an affectionate one. Both age into adulthood beautifully.