Networks are crucial components of many sectors, including telecommunications, healthcare, finance, energy, and transportation.The information carried in such networks often contains sensitive user data, like location data for commuters and packet data for online users. Therefore, when considering data release for networks, one must ensure that data release mechanisms do not leak information about individuals, quantified in a precise mathematical sense. Differential Privacy (DP) is the widely accepted, formal, state-of-the-art technique, which has found use in a variety of real-life settings including the 2020 U.S. Census, Apple users' device data, or Google's location data. Yet, the use of DP comes with new challenges, as the noise added for privacy introduces inaccuracies or biases and further, DP techniques can also distribute these biases disproportionately across different populations, inducing fairness issues. The goal of this paper is to characterize the impact of DP on bias and unfairness in the context of releasing information about networks, taking a departure from previous work which has studied these effects in the context of private population counts release (such as in the U.S. Census). To this end, we consider a network release problem where the network structure is known to all, but the weights on edges must be released privately. We consider the impact of this private release on a simple downstream decision-making task run by a third-party, which is to find the shortest path between any two pairs of nodes and recommend the best route to users. This setting is of highly practical relevance, mirroring scenarios in transportation networks, where preserving privacy while providing accurate routing information is crucial. Our work provides theoretical foundations and empirical evidence into the bias and unfairness arising due to privacy in these networked decision problems.