Why do some people omit Hashem's name when reciting Tefilat Haderech, and when is the best time to say this beracha during the journey?
With Hashem's Name or Without
The Shulhan Aruch says in Siman 110 Se'if 4 that if a person is embarking on a journey he needs to say Tefilat Haderech. And in Se'if 7 it says that a person should only say this beracha if traveling a distance of at least a parsa, but if traveling less than a parsa they should say the beracha but omit Hashem's name in the ending.
There is a disagreement among the Aharonim on what is considered a parsa. Some say it is a physical distance of 3 kilometers and 840 meters, while others say that it is measured by time of 72 minutes. Sedei Hemed in Siman 2 says that a parsa is measured by physical distance and not time. On the other hand, the book Zichron Yehuda in Siman 42 says that it is measured by time, and therefore you have only traveled a parsa if the journey will take more than 72 minutes in your mode of transportation. This is also the conclusion of Rav Ovadia Yosef in his book Yabia Omer Volume 1 Siman 13.
Besides this discussion, even if we all agree that it is a chronological distance and not a physical distance, there are still reasons to question whether it should be said. The Rabbis established this blessing specifically for the danger of highway robbers (Tehilah Lidavid) and these days highway robbers are not so common therefore many hold that today we should say the beracha but omit the name of Hashem. Furthermore, there are not many parsa length roads with no villages along the way, and there are usually police and cabins along the way. Even if one cannot see a hint of civilization, there might be someone within a parsa of their location and the Shulhan Aruch says that one cannot say Tefilat Haderech if one is within a parsa of a city.
Therefore, many have the custom to say the beracha while omitting the name of Hashem, based on the principle of safek berachot lehakel.
On the other hand, many have the custom to say this beracha with the name of Hashem because they say that when the Rabbis established the beracha they included all types of dangerous situations, not just highway robbers. Since auto accidents are a common danger, it does not matter whether there are highway robbers or civilized villages nearby. This is also why the Mishna Berura says in Se'if Katan 30 that when a person is on a train he needs to recite Tefilat Haderech if the trip is at least one physical parsa.
The above-mentioned Rabbis are lenient due to what is mentioned in Penei Yehoshua Masechet Brachot 29b which says that since Tefilat Haderech is a prayer for mercy, one can say it even if there is a doubt as to whether you should say it. Therefore, one can say Tefilat Haderech using the name of Hashem.
Each person should follow their Rabbi since there is a good reason for each side.
When to Say the Beracha
The proper time to say Tefilat Haderech is when one starts his journey, as the Shulhan Aruch implies in Siman 110 Se'if 7 and also in Se'if 6 where he quotes Rabbi Rottenburg. This is also what the Taz clearly says, that as soon as one is ready for the journey he can say it, even before getting on his way. And especially in our days where it would be distracting for a driver to say it during the journey when he is unable to pull over to say this beracha, as the Shulchan Aruch says lechat’hila one should pull over to say this beracha.
On the other hand, the Magen Avraham says in se'if katan 14 and the Kaf HaHaim says in se'if katan 52 that one must leave the area of the city before saying Tefilat Haderech. However, there is a possibility that they are referring to the dangerous situation of highway robbers in those days. Today, auto accidents happen in the city too so perhaps they would all agree that one should say Tefilat Haderech before leaving the city.
Either way, everyone should follow his Hacham and his custom.