The difference in construction explains most of their divergences. The Conference Board asks about jobs, so it tracks the unemployment-rate closely. Michigan asks about buying conditions and prices, so it tracks petrol prices and inflation.
Neither predicts consumer spending well. People report feeling terrible and then keep spending. The useful pieces are narrower: the Conference Board's jobs-plentiful minus jobs-hard-to-get spread is a decent labour market gauge, and Michigan's inflation expectations series is cited by policymakers.
Example: Michigan sentiment falls to 63 while retail sales control group growth stays at 0.5% a month. Sentiment is tracking petrol prices; spending is tracking income, and income is what actually pays for the spending.
Related: inflation-expectations, retail-sales-control-group, unemployment-rate, leading-indicator, u6-unemployment