How do block explorers interface with a chain differently than full nodes? Would it be possible to integrate block explorer features into a full node implementation in the future?
Block explorers run instrumented EVMs, as opposed to the vanilla EVM that is run by a full node. If you want the same level of detail yourself, then you'll need to know how to create and instrument your own EVM.
A good place to start - which includes links to the relevant areas of code - would be this previous thread: Instrumenting EVM
Would it be possible to integrate block explorer features into a full node implementation in the future?
Yes that would be definitely possible. It is not unlikely that some ethereum node implementations will add explorer features in near future. It would also be possible to run your own block explorer locally besides the node (https://github.com/etherparty/explorer). A good chunk of the block explorer functionality can be drawn from the RPC interface of a node. Also, the interface will be more extensive in the future, for example to allow receiving information on contract internal transactions. Parity has recently added a feature to trace these messages (https://github.com/ethcore/parity/wiki/JSONRPC-trace-module).