Let X be uniformly distributed on [-1,1]. Let Y=X^2.
Then, they are not independent:
probability that X>0.9 and Y<0.5 is 0, but prob X>0.9 is 0.1, prob Y<0.5 is something close to .7 or so, product is .07 != 0.
However, they are uncorrelated (i.e. have 0 covariance):
COV(X,Y) = E((X - E(X))(Y-E(Y)) = E((X - 0)(X^2 - 1/2))
= E(X^3 - 1/2X) = 0.