Why is there an h after the r?
heard as rytidiadelphus
spelled rhytidiadelphus
Because Greek rho at the start of a word carries a breathing mark, which Latin wrote as an h. So rytidiadelphus gains one letter and becomes rhytidiadelphus.
What the ear did
No English speaker pronounces that h, and no English speaker ever did. Greek marked an initial rho with a rough breathing, Latin wrote the mark as an h, and botany inherited the pair of letters wholesale: rhizome, rhododendron, rhubarb, rhinoceros, rhyolite. The same thing happens inside a word after a vowel, which is where catarrh and myrrh get their double r and silent h. None of it is audible in English, and none of it is optional in the spelling. For a bryologist the rule is worth carrying around, because rh opens an unusual number of moss and liverwort genera: Rhytidiadelphus, Rhytidium, Rhizomnium, Rhodobryum. If you have heard a name that starts with a plain r sound and you know it is Greek, write the h and check afterwards.
R-H-Y-T-I-D-I-A-D-E-L-P-H-U-S
The full entry for rhytidiadelphus covers the name itself.