Edging options for newly paved asphalt driveway and parking lot
Unless your neighbor pulls up his timbers to check their condition, it's impossible to say if they're truly holding up well; wood in the ground will rot from the bottom up, so the top can look great but underneath could be a soggy mess. 

That said, ground contact-rated, pressure-treated lumber (typically pine here) should last 10 years without issue, especially if you wait long enough to use it that you water seal it or stain with a moisture-sealing stain. Alternatively, you could pay more (3x or more) for a wood that is naturally moisture-resistant like cedar or redwood.

However, while I'm not very familiar with the behavior of asphalt, I do know it will flex a lot more than brick pavers, so the wood lining your driveway may well crack over time due to that.