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Ah, the best comparison I can give you between Direct Peering and Cloud Interconnect can be found on this documentation page, where Google describes how they are similar and how they are different. Basically, (and here’s another good documentation page) Direct Peering is for when your internet-connected network (i.e. you have a Publicly routable ASN) are exchanging data with Google that is not necessarily GCP-specific. Cloud Interconnect is basically for when you want to connect your network to a VPC in GCP.
As for the costs and hardware and such, the second link above notes that, peering occurs at common physical locations and both Google and any peering network bear their own costs in reaching any such location. All Google’s peering locations are in its PeeringDB entry.
And this page notes that For Dedicated Interconnect, you provision a cross connect between the Google network and your own router in a common location.