If the degree distribution of a network conforms to the long-tail distribution, but it is not a scale-free network, what properties does it have?

Question

For a real network, I plotted its degree distribution and found it conforms to the long-tail distribution. It means that most of the nodes in the network are connected to only a few nodes, and there are very few nodes that have a much higher degree than most other nodes. However, when I checked whether it belongs to a scale-free network (using powerlaw package in R language), the results showed it doesn't belong to a scale-free network. Thus, what properties can I get? Can I say that this network is not vulnerable to random attacks at nodes?