I can't even tell you what fun I had answering your question! I love this play and know most of it by heart, but never really explored these two characters as a pair. What an interesting set of characters to compare! There are many more differences than similarities in these two characters, so let me begin with the similarities and end with the all-important differences.
Both of these women want to help Helen and serve as a mother-figure in some way. Yes, it is redundant to say that the two are both women, but that is the first (and the most obvious similarity). They also both serve as a mother figure in some way. Kate, of course, is Helen's real mom. Annie, of course, is Helen's teacher, but especially when she spends a few months apart from the family in the small cottage adjoining the property, Helen has to learn to depend on Annie for the necessities of life. The most important similarity that both women share is the desire to help Helen. I will go even further and say that it is an INTENSE desire to help Helen. This is where the similarities end and the differences begin. The differences arise in exactly WHY and HOW they intend to help Helen.
The first difference Kate and Annie have is a difference in their pasts. Kate had an easy past with "old money" and riches. Kate is the wife of a very prominent "Captain Keller." They live on a vast plantation with many servants and much land. There is nothing to indicate the Kellers are a rags-to-riches story. They are simply the outcome of society in the old South. This life of luxury in the old South unfortunately led to a weak woman. Kate, quite simply, is ruled by her husband. Luckily, she is wise enough to know how to get her way (as in hiring Annie). Annie, on the other hand, had an extremely difficult past full of poverty and illness. She was blind and can only see with much difficulty because of an operation and glasses. No life of luxury for Annie, ... ever! Annie describes these effects here:
Annie. ... The youngest were in another ward to have babies they didn’t want, they started at thirteen, fourteen. They’d leave afterward, but the babies stayed and we played with them, too, though a lot of them had—sores all over from diseases you’re not supposed to talk about, but not many of them lived. The first year we had eighty, seventy died. The room Jimmie and I played in was the deadhouse, where they kept the bodies till they could dig—
Kate (closes her eyes). Oh, my dear—
Annie. —the graves. (She is immune to KATE’S compassion.) No, it made me strong. But I don’t think you need send Helen there. She’s strong enough. (She waits again; but when neither offers her a word, she simply concludes.) No, I have no conditions, Captain Keller.
The next difference has to do with WHY the two women want to help Helen. Kate wants to help Helen because Kate has PITY on Helen. Of course, Helen is Kate's own child and, as such, has much compassion due to her (thinks Kate).
[HELEN is back at AUNT EV, fingering her dress, and yanks two buttons from it.]
Aunt Ev. Helen! My buttons.
[HELEN pushes the buttons into the doll’s face. KATE now sees, comes swiftly to kneel, lifts HELEN’S hand to her own eyes in question.]
Kate. Eyes? (HELEN nods energetically.) She wants the doll to have eyes.
Watching the unruly Helen live in her own world of darkness is a sadness for Kate. She wishes to have peace in the household and a child who will not harm the little ones nor disturb the older ones. Standing as a complete opposite to Kate here is Annie who wants to help Helen because Annie wants to prove her success as a teacher. This doesn't necessarily mean that Annie doesn't "like" Helen or "feel for" Helen, but that is not her main impetus. Annie, due to her situation in life, is forced to make a way for herself and, due to her disability, must teach others who are in the same situation as herself. Before this time, Annie was at a collective school and this is her very first position. Her failure at this position may nix her career entirely.
Perhaps an even more important difference is in HOW the women plan to help Helen. Kate has two methods: coddle and ask for help. Thank God for the latter! Coddling Helen, due to Kate's pity, has made Helen into a monster. Annie says so herself.
Annie. I know an ordinary tantrum well enough, when I see one, and a badly spoiled child—
James. Hear, hear.
Keller (very annoyed). Miss Sullivan! You would have more understanding of your pupil if you had some pity in you. Now kindly do as I—
Annie. Pity? (She releases HELEN to turn equally annoyed on KELLER across the table; instantly HELEN scrambles up and dives at ANNIE’S plate. This time ANNIE intercepts her by pouncing on her wrists like a hawk, and her temper boils.) For this tyrant? The whole house turns on her whims. Is there anything she wants she doesn’t get? I’ll tell you what I pity, that the sun won’t rise and set for her all her life, and every day you’re telling her it will. What good will your pity do her when you’re under the strawberries, Captain Keller?
Luckily, Kate does have the gumption to ask for help for Helen and makes sure her husband acquires that help for the family. In regards to Annie, the HOW of helping Helen has to do with being stubborn (and often downright abrasive) with her own tried and true teaching tactics. Although the above quote serves as a perfect support for this argument, there is another near the conclusion of the play:
[KELLER shoots a glare at him, as HELEN plunges her other hand into ANNIE’S plate. ANNIE at once moves in to grasp her wrist, and HELEN, flinging out a hand, encounters the pitcher; she swings with it at ANNIE; ANNIE, falling back, blocks it with an elbow, but the water flies over her dress. ANNIE gets her breath, then snatches the pitcher away in one hand, hoists HELEN up bodily under the other arm, and starts to carry her out, kicking. KELLER stands.]
Annie (savagely polite). Don’t get up!
Keller. Where are you going?
Annie. Don’t smooth anything else out for me, don’t interfere in any way! I treat her like a seeing child because I ask her to see, I expect her to see, don’t undo what I do!
Keller. Where are you taking her?
Annie. To make her fill this pitcher again!
The irony is, it is precisely HERE that Helen has her breakthrough and learns that every word has meaning. Annie's stubbornness and grit pay off. Helen learns, ... and eventually hands "the key" of learning to "teacher."
I guess you could say, in conclusion, that Kate and Annie serve as foils to one another where one remains static (Kate) while the other is dynamic (Annie). Kate continues with her compassion and pity for Helen. I can't forget the actress when she finally welcomes Helen back into her arms after the months apart. Kate, there, doesn't give a care what has happened, she just wants her child back. Annie, however, goes from a stubborn teacher who doesn't care about Helen to a dedicated mentor who cares VERY MUCH about Helen. In fact, Annie cares SO much that the two spend the rest of their lives together having many more adventures.