Your two sentences are equivalent, but the version using 'would' is more formal. 'used to' and 'would' are always interchangeable when you are talking about a past action that was repeated. Using 'would' makes the sentence a little more formal. In your example, your mother took you to school many times. An action that was repeated many times and happened in the past, so you can use either 'used to' or 'would'. 'used to' and 'would' can mean other things besides a repeated past action (like your example). In these cases, they are not interchangeable. For example, 'used to' can describe a past state that is no longer true. "I used to live in France". We are talking about a past state and not a past repeated action/activity, so you cannot use 'would' here. "I used to be always sick, but now I'm fine". Again, we are talking about a past state (not repeated action) that is no longer true. We cannot use 'would' here.

Habit in the past. [simple present] When I was young, my mother took me to school. [modal would] When I was young, my mother would take me to school. [semi-model used to] When I was young, my mother used to take me to school [but she doesn't anymore] Used to (do) shows a contrast between the past and the present. The condition is no longer true. I used to go to the movies every Saturday. [but not now] I didn't use to read books. [but now I do] I recommend the famous grammar book by Murphy "English Grammar In Use." See units 18 and 36C.

I think that this phase is a routine, maybe you should use present tense. "My mun take me to school every day" Every day is a routine, but I'm not a teacher.. But if you're talking about the past is correct your sentences, I think so

Thank you, but the book claims there is only one possible answer (as always..)

I think the overall meaning is pretty much the same but "would" is a bit more formal