LON, the nails are inserted in the first surgery and then locked in place when frames come off. Not as invasive second time around, so the patient can walk (assisted) sooner after surgery with less pain.
LATN, after lengthening you need a few weeks for correction using a hexapod. After that, nails are inserted and locked in second surgery when frames come off. So there's more pain involved on the second round. Luckily bone healing speeds up at this stage for LATN patients.
Benefit of LATN is outlined on
page I linked to earlier:
A large advantage of LATN is that bone formation is fast and because of strong internal fixation, compared to LON, patients can return to normal activity 2 or more months faster. Also, since it undergoes a correction process, precise bone alignment can be met.
And here's another comparison, again
from link above:
There may be several advantages of the LATN over the lengthening over a nail technique, including the ability to insert a full-length large-diameter nail for more stability. In the lengthening over a nail technique a small-diameter nail is used so the bone can slide over the nail. In addition the nail is pulled out of the distal fragment with lengthening leading to suboptimal stability; also, distraction over a nail can sometimes be difficult and mechanical binding from nail impingement can prevent lengthening [18]. This problem is avoided with the LATN technique. With the LATN technique, there is no concomitant use of internal and external fixation, and this should lower the risk of infection. In the lengthening over a nail procedure, if a pin tract infection occurs during the distraction phase, the presence of the nail increases the chances that the infection will spread to become a deep infection. If a pin tract infection develops during the LATN technique, it can be treated with antibiotics or with pin removal prior to IM nail insertion. Additionally, the timing of intramedullary nail insertion can be adjusted if need be. Like the classic method, LATN affords the ability to gradually correct diaphyseal deformity and lengthen prior to nail insertion. Lengthening over a nail or use of an internal lengthening nail requires an acute deformity correction which compromises bone healing potential during subsequent lengthening. A distal third tibial deformity could not be treated with lengthening over a nail since the IM nail would be pulled out of the distal segment during lengthening.
TL;DR of above quote: compared to LON, LATN allows for stronger nails for stability, less mechanical resistance during lengthening, infections are less serious, and allows for deformity correction.