We consider a simplified model of a social network in which individuals have one of two opinions (called 0 and 1) and their opinions and the network connections coevolve. Edges are picked at random. If the two connected individuals hold different opinions then, with probability 1 - alpha, one imitates the opinion of the other; otherwise (i.e., with probability alpha), the link between them is broken and one of them makes a new connection to an individual chosen at random (i) from those with the same opinion or (ii) from the network as a whole. The evolution of the system stops when there are no longer any discordant edges connecting individuals with different opinions. Letting rho be the fraction of voters holding the minority opinion after the evolution stops, we are interested in how rho depends on alpha and the initial fraction u of voters with opinion 1. In case (i), there is a critical value alpha(c) which does not depend on u, with rho approximate to u for alpha > alpha(c) and rho approximate to 0 for alpha < alpha(c). In case (ii), the transition point alpha(c)(u) depends on the initial density u. For a > alpha(c)(u), rho approximate to u, but for alpha < alpha(c)(u), we have rho(alpha,u) = rho(alpha, 1/2). Using simulations and approximate calculations, we explain why these two nearly identical models have such dramatically different phase transitions.