Pretty important. I've had tons of problems with my old gateway, and on top of that, the problems dealing with their tech support made me realize that's a brand I'm going to avoid in the future. HP has its own sets of problems, but their tech support at least is great as far as I've heard. Gateway wanted me to pay extra for a reformat disk that they should have given out for free in the first place instead of their flawed built-in recovery logic, which by the way, can be destroyed by viruses, which is something I learned from experience. Also it took a couple days for me to finally get them to send the reformat disk because their tech support people wouldn't listen to my explanation, barely spoke english, and probably had no idea what they were talking about in the first place.
Well if we're talking about specs that are exactly the same, then we should also look at the peripherals such as quality of speakers, user friendliness of the keyboard, programmes that comes specific with that certain brand, design and one frequently overlooked point is the comfort level the product gives while one uses it. Make sure it is something you'll look forward to using rather than something you're using because it is a necessity. When we're done with that, go on to see how each company's product compares with each other in terms of durability and reliability. After-sales services are also important in your decision making process. You wouldn't want to pay for technical support for as long as possible since computers often run into problems nowadays regardless of their brand!
Keep in mind for every person on this board complaining about particular brands there are just as many people who are happy with them. My Dell didn't last as long as I would like but my roommate still has his 8 years later. I have a Gateway I bought in 2003 that got struck by lightning and still works perfectly. I have a Toshiba now that has been going strong for over 4 years. I think the brand has very little to do with the make up of the actual computer.
I don't think the brand of who put the computer together matters but the brand of the parts matters. For example I would always want my computers to have a Intel processor over an AMD processor. I always want my computers to have a Nvidia graphics card over any other graphics card. Speaker brand can also matter a lot too. I have an ASUS computer with Bang and Olufsen speakers and they are MUCH better than Beats speakers. The only reason why the actual computer brand would matter is for the customer service quality.
@nick87 You might not want to jump on that HP support wagon too fast. While better then what you report, there are still a lot of issues regarding how they do things and how well the experience overall ends up being. That said everyone I talked to spoke english well and wasn't someone I'd like to stab. So yeah, there is that.