Note that I'm basing my answer on largely on this site and this site, both of which use the Drik Panchangam, so that might bias the arguments somewhat.
The Vakya Panchgangam system, as used for instance in the famous Pambu Panchangam and the Sri Rangam Panchangam, originates in an ancient astrological work called the Surya Siddhanta. The Surya Siddhanta covers many topics in astrology, but most notably it has a section with a collection of simple mnemonic devices that describe how to determine the location of the heavenly bodies like the Sun and Moon, based on trigonometric calculations that were quite advanced for its time. These mnemonic devices, written in the form of simple sentences with easy-to-remember words (Vakya), were extremely accurate in describing the motions of heavenly bodies at the time of composition. But the author knew that they would only be accurate for a very short amount of time, so he wrote that these Vakyas should be regularly updated by observing the actual positions of the heavenly bodies using the human eye and then modifying the Vakyas to match the latest motions of those bodies.
And accordingly future generations kept rewriting the Vakyas, but unfortunately the last person to make an update was the Indian astronomer Bhaskaracharya nearly a thousand years ago. So practitioners of the Vakya Panchangam are relying on astronomical calculations that are incredibly out of date. That is why people often use Drik Ganitha Panchangam, AKA Thiruganitha Panchgam, where you don't rely on the Vakyas at all, and instead rely on the actual positions of the the heavenly bodies. Drik comes from the Sankrit word "drishti" meaning sight, because it originally relied on naked-eye observations, but Drik-Panchangam websites today rely upon methods that are even more precise than the naked eye, like sophisticated modern astronomical calculation techniques, as well as Ephemeris data from NASA.
Yet there are lots of people that still rely on Vakya Panchangam. Why? Well, it's tradition to use the Vakyas, as the Surya Siddhanta is considered divine. But the extent to which the Surya Siddhanta is out of date was demonstrated by the 1950 solar and lunar eclipses, where the Vakya Panchangams published dates and times for the eclipses that were dramatically different from what actually occurred. From that time onwards even the publishers of Vakya Panchagams have started using certain astronomical data, like eclipse times taken from Drik Panchangam publications.
So if you want the most accurate information out there, it's advisable to use modern Drik Panchangams found on the Internet, but there is the downside that you're not keeping with the old traditions. What would be ideal is if someone today were to write updated Vakyas. People up to the time of Bhaskara were willing to do that, but in the modern day people are too scared to make changes to the Sanskrit words of the illustrious writers of the past. But that concern may be unfounded; we have excellent Sanskrit writers living today, like Rambhadracharya whose work has been compared favorably to that of Kalidasa. I think someone like that may be qualified to rewrite the Vakyas of the Surya Siddhanta, if they only have the courage to do so.
And to answer your other question, this site says "Maharishi Vashistha and Varahamihira have said to make the panchangam as per Druka ganita." So presumably that's what Vashishta used for Rama's horoscope.