Both for me, depending on who I'm talking to at the moment. I'm familiar with many Latin names due to having a plant sales biz so it's a necessity. Many customers still rely on common names so it's just as important to learn what they are so I can help them out. Plus, some common names are fun! And yep, as someone above said, some common names refer to one plant in an area but to a completely different plant in another area/state. Might as well continue to grasp them all for future knowledge (even though I'm often saying, "hmmm...that sounds familiar to me but can't place it at the time."
Shoe -off to happily broadcast some Festuca arundinacea (tall Fescue); grateful I'm not spending my time pulling Cynodon dactylon (Bermuda grass a.k.a. a whole host of other common names not acceptable for the public eye)