SA-516-70 is commonly used for CS fabrication. It has a higher allowable stress in ASME VIII-1, allowing fabricators to design thinner pressure boundaries.
SA-516-60 is a nominally weaker material, but it comes with the benefit of typically being 'softer', which lends itself to certain applications. For sour service for example, hardness is an important variable in the design. By specifying SA-516-60, you are typically lowering the material hardness, making it less susceptible to sour service attack.
If your customer has specified SA-516-60, you need to understand why. If the vessel is in sour service, you may have a hard time convincing them to accept a substitution.