I have a different surname to my dc. When I travel with them, I carry a standard form which says that dh gives consent for us to travel. No-one has ever asked to see it. The closest I got was the passport control asking ds2, "is this lady your mum?" to which he said, "yes". I said, "I have birth certificates and a letter from their dad", but she just said, "that's OK. You have two young people here who are much more reliable at identifying you".
I choose not to take my dh's name. Other people choose to do so. Either way is fine. What I don't like is people who assume that I am Mrs. DHname without asking (in any case, we're both Dr., so it would be Dr. DHname and Dr. DHname, like Bounty and that would get very confusing!).