2) The Participation Story. You aren't likely to hear the administration mention the word "participation" when it talks about this report. That's because the second milestone of this morning's report is that the labor force participation rate -- the share of the population working or looking for work -- fell to to 63.6 percent, the lowest level since 1981. If more adults were playing along, the unemployment rate would almost certainly be much higher than 8.1%. (That's why you sometimes hear conservatives talk about the "real" unemployment rate when they factor in this fall-off.) Here's the 60 year perspective, from the great Scott Barber:

It's reasonable to think that some of this fall-off is driven by the recession. But it's also driven by demographics. America is getting older. The 65+ population will increase by 80 percent this decade, no matter what comes out of the BLS. My guess is that long-term unemployment, a scourge which is concentrated among elderly workers, is pushing some into early retirement.
3) The Temperature Story. "Nice weather we're having," is the
kind of thing one party says to another party with which they share
nothing in common. So it's fitting that both sides of the aisle can
agree that good weather in the winter probably acted like a stimulus by
pulling forward some hiring and spending on homes. As a result, no matter what you
think about the economy or Obama, it's pretty clear that America's
"early spring" in the winter might have stolen some economic activity
from the actual spring. That's a reasonable explanation for why job
creation has fallen by about half from December/January/February to
March/April.
4) The "Long" View. I'm putting two facts in my "long" view category. The first is long-long-term unemployment, the ultimate tragedy of the Great Recession. It appears to be mitigating. In April, 3.7 million Americans had been jobless for a year or more. That's horrible. But it's 600,000 better than the 4.3 million long-long-term jobless we had in April 2010.
The second long view comes from the Hamilton Project, which built a calculator to determine when the U.S. would finally dig ourselves out of the "jobs gap." If we're experiencing a blip, and winter was the new normal, our rate of job growth will have erased the 12-million worker gap before the end of the decade. But if April is indicative of the new normal, and winter was a blip, we won't erase the gap until the 2030s. Spin today's numbers how you'd like, but that's a future nobody should hope for.