When Hagar flees before Ishmael is born, she is referred to as a שִׁפְחָ֥ה . When she and Ishmael are expelled together, she is a אָמָ֖ה.
What is the difference between these two kinds of female servant?
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The first difference is the level of status and the resulting work they do. The Kesav VeHakabbalah in Parshas Mishpatim 21:7 notes:
לאמה. המשרתת תקרא אמה, והיא לרש"פ מדרגה חשובה מן השפחה, שהשפחה עלי' מוטל כל עבודות הקשות והפחותות שבבית (מאגד), והאמה אין עלי' רק עבודות הקלות וההשגחה על סדור ותקון הבית
As a maidservant - Someone in service is called a maidservant (amah) and it is a level more important than that of a shifchah, since a slave girl (i.e. a shifchah) is assigned all the difficult and menial chores in the house, whereas the maidservant (i.e. the amah) only has to do the light work and oversee the order and repair of the house.
Which in the context of your question makes sense as Hagar, the daughter of Pharoah was initially given over to Avraham as a servant following the miracles that happened for Sarah when she was in his custody. Bereishis Rabbah 45:1 famously concludes there that Pharoah reasoned:
אָמַר מוּטָב שֶׁתְּהֵא בִּתִּי שִׁפְחָה בְּבַיִת זֶה וְלֹא גְבִירָה בְּבַיִת אַחֵר
As he (Pharoah) said, "It is better that my daughter be a servant in (Avraham's) this house than a mistress in another house."
However, when she was sent away, she was Avraham's wife and so arguably, her level of work may have been lessened and therefore it is fit to call her an אמה.
Alternatively, the Netziv on Parshas Lech Lecha 16:2 writes:
בא נא אל שפחתי. לא כדבר רחל ליעקב הנה אמתי בלהה. דאמה משמע לא כשפחה ממש שהיא קניית הגוף אלא סרה למשמעתה. כמו אמה העבריה והיינו דאע״ג שלבן נתן אותה לשפחה מ״מ מאז מסרה רחל אותה ליעקב היתה בתורת אמה. משא״כ שרה התנהגה גם אח״כ בתורת שפחה כמו שיבואר
Consort with my maid - Not like Rochel said to Yaakov, "Here is my maidservant (אמתי) Bilhah". Since the implication of "Amah" is not like "Shifchah" exactly, which is the purchase of her body, but rather to turn her over for absolute obedience, like a Hebrew slave (אמה העבריה) and that's why even though Lavan gave her over as a shifchah, nevertheless from then Rochel gave her over to Yaakov with the laws of an amah. Whereas conversely, Sarah treated her (Hagar) even afterwards as a shifcha, like we explained.
אָמָה usually refers to a Jewish girl who was sold into slavery which she can only be until the age of twelve. (אָמָה הָעִבְרִיָּה)
שִׁפְחָ֥ה usually refers to a non-Jewish woman (can be any age) who in slavery (שִׁפְחָה כְּנַעֲנִית)
The Seforono says she is called שִׁפְחָ֥ה before Ishmael is born because she had not yet been freed at that time. Possibly she is called an אָמָה later on because she had some degree of freedom at that point in time. At least enough not to refer to her as a שִׁפְחָ֥ה anymore.
However, there must be other reasons as well because by Bilha and Zilpha the Torah also alternates between calling them שִׁפְחָ֥ה or אָמָה. They continue to be referred to as שִׁפְחָ֥ה even after being called אָמָה. (They are also referred to as wives and other titles earlier and later on. The reasons behind the some of those varying titles have explanations given by the basic commentaries on Chumash)
Rav Yaakov Kamenitsky who was a major expert on dikduk writes on these pesukim about Hagar similar to what I write above that the question asked in the OP but concludes that this needs further research. He also points out that although in the Mishna and Gemora אָמָה always means Jewish girl in slavery and שִׁפְחָ֥ה always means non-Jewish women in slavery there is one place in Chumash where אָמָה refers to non-Jewish women slaves so apparently there is further nuance in when and why these words are used.