They are the same word: orthopedic is American English and orthopaedic is British English, both covering the treatment of bones, joints and muscles. If you are writing for a UK reader, keep the extra a in orthopaedic, the way you would in paediatric.
The core difference
Orthopedic and orthopaedic are identical in meaning; only the spelling changes depending on American or British English.
- orthopedic — American spelling, relating to treating bones, joints and muscles: He wears orthopedic shoes to support his flat feet.
- orthopaedic — British spelling of the same word, also used for supportive products: He sleeps far better since switching to an orthopaedic mattress.
How to tell them apart
There is no meaning difference to remember here, only a spelling one. British English keeps the ae from the original Greek and Latin, the same pattern you see in paediatric and aesthetic, while American English simplifies it to a plain e. If in doubt, match whichever variety of English the rest of your document uses.
Both words trace back to the Greek orthos, meaning straight or correct, combined with a root referring to a child, since the field began with correcting skeletal problems in children. That history is shared, so it will not help you pick a spelling; the audience will.