Asavil,
It is my opinion that having a Green Card, can definatley influence you in many ways. At the very start, there are many programs out there that simply DO NOT sponsor any visas. When it comes time to start applying, you'll find many programs that you might wish to apply for, but you simply CAN NOT, because of your Green Card status, or lack of. Simply put, that to me, makes a huge difference.
Visa sponsorship by not doubt is looked upon as a burden by all programs. Regardless of how often the programs sponsor a visa, there are still many unpredicatble outcomes of the entire visa process. Some of which could delay your paperwork, and hence delay your start date.
Simply put, if you have the opportunity to get a Green Card then go for it.