I am getting a new shingle roof to replace a slate roof. Would a new shingle roof need a different underlayment that the slate roof does not have?
Not from what I've seen. From the dozen of so slate roofs I've seen and had, they were all applied directly to flat tight seamed planking. Typically, only wood or other types of roofing that aren't waterproof and need to dry-out have a furring-strip or stepped off underlayment.
You'll just want to replace any rotted areas with new planking (best) or thickness matching plywood (i.e. 3/4" or 2-layers of 1/2"). Tar paper would go on top of that on a problem-free roof. If the eave decking is all rotted and therefore has been leaking, then you'll want Ice and Water underlayment over the eaves.