The difference between the two lies, unfortunately, in a concept rarely taught in American English classes.
"We walked" = preterite (simple past)
The preterite indicates an action that occurred in the past and has since been discontinued.
"We would walk" = conditional present
This verb tense indicates an action that began in the past and is possibly ongoing at the present time.
Employing the conditional present brings a flavor of reminiscence to a story as opposed to a feeling of recitation of the facts that the simple past brings.
Edwin Ashworth stated it well; I just used more formal labels.