Hi Miriam,
Thanks for the response!
So the only way the two business officially differentiate themselves is by the year they were established on their business licenses. For example, we're "ABC Company 1983 Ltd." and they are "ABC Company 1995 Ltd.". We tried changing the Google+ page to "ABC Company 1983 Ltd." and building citations with that name, but the problem is people only search for "ABC Company" and we're pretty behind compared to them in citations, anchor text links, etc. I guess the only thing we can do is keep our name without the date, and just out-build them. Thanks!