listening in the sens of putting your head on someone chest is écouter- this apply if your child do so with no real objectives. If your doctor perform the same with the objectives of looking if everything is fine, the he/she will ausculte you. Ausculter has a medical connotation and it is about checking someone out. Hope this clarifies

ausculter is écouter when it is done by doctors when they examine you. They can ausculter le cœur or ausculter le patient meaning they examine the patient (by listening to their organs).

• Le docteur m’ a écoutée : The doctor listened to me, to what I had to say
• Le docteur m’a auscultée : The doctor examined me

You can’t use ausculter with an infinitive as you can écouter. ausculter le cœur battre is not possible in French.

The medical use goes back to the early 19th century and to a famous French doctor, Laennec, who invented the stethoscope. The word quickly spread from French to other languages and started being used in this specific meaning by medical doctors.

The difference, as I see it , is that in the act termed “ausculter” you listen with a professionally trained ear, whereas when you are given to the action that is “écouter” in the general case you just take in the sounds as they come ; this is not an absolutely perfect criterion of differentiation of the terms, as when you listen for instance to music (écouter de la musique) you do do that in the context of a musically trained ear, but the outcome of this comparison to musical standards, even for a professional listener such as a music critic, is far from the very pragmatic act of the doctors.