Both mean they same thing. “She is with child” is not as common. Saying “She is pregnant” is a lot more common.

“She is with child” A very old-fashioned way of saying someone is pregnant. Someone would smile and laugh a little at it I think. It’s correct, just not very current. These days, “she is pregnant” or “she’s expecting” is more common. “She is having a baby” can have two meanings. 1. “She is pregnant.” 2. “She is giving birth right now!!” Although in this case it would be more common to hear “she is having THE baby” if she was giving birth now:)

"She is with child" sounds poetic. Say it if you want to use colorful language. "Having a baby" can mean either actually delivering, as in a hospital, or just being pregnant. If someone says to you "Have you heard that Jane is having a baby?" it does not mean Jane is in the hospital delivering. It means Jane is pregnant.

"She is with child" is very old-fashioned. It is something you read in old books. It is language from the 1800s or earlier. "She is pregnant" is plain and factual. "She is expecting" is a softened, more polite way to say "she is pregnant." "She is having a baby" can just mean "she is pregnant." It often carries the idea that the birth will be pretty soon. If someone says "She is having a baby," a natural response might be "Oooh! When?" "Where is Mary?" "She's in the hospital--" "Oh, no!" "No, no, it's something good! She's having a baby!"

In my perspective, "She is with child" - means she is pregnant "She is having a baby"- would also mean, right at this moment she is delivering the baby