How can I copy specific tables into one SQL Server Table into a separate table automatically?
The instant answer that springs to mind is transactional replication. If it needs to be bi-directional, then peer-to-peer transactional replication.

If the latency here proves too high, or if the schemabinding is too different, then you could have a synchronous solution, with a linked server between the two DB servers and a trigger on the table which fires on insert/update/delete, that updates the table on the other server using the linked server. Warning here is that triggers can degrade performance a little.