There are likely others with more insight, but my impression for the younger teams (e.g., 2/3/4 grade) is that the BLC and NL boys club teams are pretty similar. Both teams have similar practice schedules and play in roughly similar leagues and tournaments. They also play against each other often. I would say that the BLC teams are, on average, somewhat better than the NL teams. However, there is variation year-to-year in that there are individual grade years where one or the other is stronger, so it's hard to make a generalization that would be true for all grade years. NL practices at Landon and WES fields, so it's probably more convenient for someone who lives closer in Bethesda and Chevy Chase or is in DC. I think BLC practices tend to be slightly further out in Bethesda and in Potomac, but not sure. Coaching is fairly comparable though that also depends year-to-year since a specific NL or BLC team might have a particularly good coach. The brothers that run NL are both very nice guys and I think especially good with the younger kids. Overall, my sense is that for the younger levels, people default to the club team with the rec league they started. So if you start in the Bethesda rec league you're more likely to play BLC and if you start in the NL rec league you're more likely to play for an NL club team. BLC probably has more kids from private and public schools slightly further out in MoCo. NL tends to draw from some of the DC privates and the close-in Bethesda public and private schools, at least for the youngest teams.
There are different considerations for the older teams, but others would probably be able to better speak to those.