Awsi nailed it exactly. ESPN is the far better option early on. They have a much wider access to college tape, better overall on-air talent and much better production values. Also, most of the pre-draft shows I watch are on NFL Network, which means I've already heard every clever line or unique angle I'm ever going to hear from those guys. I watch comparatively little of ESPN's pre draft shows, not just because they imbed them in other shows I don't want to watch, but also to keep it fresh. That's the best fit for me, as ESPN holds the nostalgia angle by a wide margin and it's overall a better viewing experience. The one year I watched NFL Network's coverage throughout was my least favorite draft to watch of any I've seen, and I've seen every pick of every draft since 1999. With that much history, all things being equal I'm going to prefer ESPN. But in this case it's not equal.
ESPN's coverage definitely falls away as the draft goes on, though. Awsi is right to point that out, with interminable interviews with coaches and GMs that waste valuable time and are bookend by commercials so that you lose 10 picks by the time they come back. By the third day I'm watching NFL Network.
Another thing: it still bothers me that they've extended it to three days and shortened the time between picks in the early rounds. It used to be a true Iron Man competition to watch it all. You had to plan out your meals and showers ahead of time. Now it's so much easier to watch it's opened the door to much more armchair analysis by people who don't spend the time. In that way it's a classic Goodell move... giving the league more surface area with no concern for depth, like pouring a glass of water on the floor.