... Why not just send it via normal mail? ...
Normal mail over what distance?
This combination by itself
gives you Paris (France) or Paris (Texas) to Connecticut in minutes, with same day delivery by post ... but we assume too much.
I suspect that what you have is the hardcopy that was sent as a courtesy after the message was delivered by phone, or as a backup because the phone call had not yet been successfully completed.
I came in at the very end of telegrams, but I remember having the option of deliver-by-hand (pricey, slower, and rare) versus deliver-by-local-phone-call (cheaper, often faster).
One deciding factor would be the venue of the target. To reach someone at home is one thing. But people did, for example, send a telegram to The Happy Couple to be read at the reception, which required hand delivery ... cheaper than a plane ticket, eh?