Hello,
First, welcome to the ACA Forums!
The TransAm is an almost 50 year old established route. It is also a somewhat "historical" route in the sense that it starts at an historical area of the USA's beginnings and has numerous places it offers the USA's history.
The route has/is well researched, really good maps with all the services a touring bicyclists needs shown so you don't have to do the research of say where to camp for the night, etc. The route offers a very diverse topography, climates, etc. regions where the "feel" of the route changes about every week or two so you do not always have the same look or feel to the route. This is especially nice since the route takes about 2.5 to 3 months to complete if you do not rush.
The big difference is that the TA is almost entirely paved. Granted the TA has much more traffic in places but overall, ACA has done a good job of avoiding it.
The GART is a "work in progress" based on a route of abandoned railroads that have been converted to trails. It too offers a diverse topography but not as much since overall it tends to follow a flatter route. Though flat is nice, this means you will not see as a diverse area since there are not many abandoned railroad tracks in mountainous areas. While there are maps, GPS info, etc. about the GART, it is definitely more in the "pioneering" stage where you may have to do a lot of route search and/or services research so you may spend much more time on tour looking for a place to camp for instance. If I had to guess, I would say the GART has less traffic compared to the unmodified TA (see below) but the TA has more services available.
To me, one big negative of rail trails (and thus the GART) is you very well may ride in a tunnel of trees. While this sounds nice, after 50 miles or so, it gets a bit boring when all the scenery is just beyond the trees. Also, as mentioned above, the TA is almost entirely paved while the GART has lots of unpaved trails, some good and some not so good. I do not know what the percentage of unpaved sections are but for instance the Cowboy Trail in Nebraska is almost entirely unpaved and the surface is not as smooth as say the Katy Trail so wider tires are definitely needed compared to the TA.
Most of the trails on the GART are nice, i.e. GAP, Ohio, and some are spectacular, i.e Hiawatha in Idaho/Montana (ok, that is a spur but it is right off the GART route) while some not so nice or the same (Cowboy trail). However, it leaves some nice ones out like the Katy and the Mickelson trails.
Unfortunately, you did not say what you are looking for, your lodging preference, etc. See
https://forums.adventurecycling.org/index.php?topic=14587.0 for info on requesting better route advice. Since we don't have that info, I guess it really depends on what you are looking for. If you want less traffic, don't mind slow going at times, and a flatter route, and don't mind doing services research then the GART is probably for you. If you don't mind some traffic (very little to moderate), a definitely more diverse route elevation profile, and have decent maps already to go, then the TA.
There are several videos out there that show the condition of various trails, i.e. GAP and Cowboy trails, to see if these are right for you. Additionally, check out CrazyGuyonaBike.com and/or Cycleblaze.com for journals of people who have ridden the GART.
As mentioned above, by using a modified TA, you could come close to combining the two. By combining ACA's Eastern Express Connector Route and/or choosing various routes within the unofficial Eastern Express Alternate Routes (
https://ridewithgps.com/collections/2244653?privacy_code=IeKurbFOSOwNV3t50ARrQwuy3dTAcWvP ) and Katy Connector Route Network (
https://ridewithgps.com/collections/1838413?privacy_code=VwkXD6IyPPcX4pFH ) and then connecting to the TA in Rush Center, KS, you can get a flatter, less trafficked TA for eastern half the TA.
If you prefer, you could go from say Platte City, MO, (Katy Connector) to the NW using various rail trails (Chief Standing Bear and Mickelson come to mind) and/or other routes (ACA's Parks, Peaks, and Prairies Route) and combine it with the GART to get a route that is ideal for you. As mentioned previously, if you provide more info on your needs/desires, the community can offer you more focused suggestions.
Whatever you choose, have a great tour! Tailwinds, John