What makes a pulley that is rated for horizontal pulling different from a pulley rated for lifting?

Most of my experience is with various blocks and pulleys on sailboats, and I can see one significant difference between the two McMaster pulleys you linked to:

The sheave of the lifting pulley has a very different cross-section than the horizontal one. It's probably assumed that the cable in the lifting pulley will always be loaded and will sit deeply in the groove. With that cross-section, if the cable goes slack it could jump out of the groove and sit on the side of the pulley.

The sheave of the horizontal pulley is more like those on sailboat blocks, a smooth groove that will self-center a line/cable as load is applied. This makes more sense is the line or cable will sometimes be loaded and sometimes be slack.

https://www.harken.com/productcategory.aspx?taxid=425

If you examine the catalog page, the majority of lift devices comply with specification ASME B30.26 (Rigging Hardware), while the horizontal do not meet any specification.

So it is easier to justify why a device is a lift device, rather than a horizontal pulling device.

American Society of Mechanical Engineers (ASME) has created the B30 Safety Standards for material handling equipment, cranes, hoists, rigging equipment, and below-the-hook lifting devices. The 30 standards cover selection, use and maintenance of equipment.

ASME rely on manufacturers to test their lift products to B30 standards. Search terms like "Meet or exceed ASME B30.26" and "Proof Tested".

Horizontal AND Lift devices are needed. But if a lift device fails, the load becomes a pendulum, hence the need for safety standards. A horizontal device fails and the load stops moving. Someone may still die or be injured, but typically not because of the load. Lift devices still fail, but ASME B30 outline safety checks to minimize failures. Occupational, Health and Safety require operators to perform these safety checks.

As for why a device is appropriate for horizontal pulling versus lifting. The horizontal device does not meet some aspect of ASME B30 or the manufacturer did not go through the ASME B30 certification process because both horizontal pulling and lifting devices are needed.