Al1943

The 9-speed Tiagra is rated for a 27 tooth maximum cog. It may be able to handle something larger, but a 34t is very doubtful. The Deore is a mountain type derailleur and can handle a 34. Your gearing combinations require a rear derailleur wrap capacity of 45 teeth (48 - 26) + (34 - 11). Depending on the exact models, the Tiagra RD is rated at 39t total capacity, and the Deore is rated at 43t. Clearly the Deore is the better choice. In fact with the Tiagra RD if you shift into the smallest chainring and smallest cassette cog I would expect the derailleur to bottom out and cause slack in the lower run of chain. And the Deore RD would probably have about all the chain it can handle.