Bridged Connection: The VM gets direct access to the network (as if it is its own device) which allows the VM to access DHCP and its gets its own public IP address. This operates on Layer 2 of the OSI model.
Can be considered less secure as the VMs IP is viewable like any other host on the network.
NAT (Network Address Translation) Connection: The VM receive its IP address via the HOST that is doing the network address translation and the IP is assigned by the DHCP running on the host. The VMs are on a different subnet. This is Layer 3 of the OSI model.
Can be considered more secure as the IP of the VM is hidden behind the host making it less directly accessible.