Tbilisi is ten administrative districts that almost nobody names, and several dozen neighbourhoods that everybody does. Listings use the neighbourhood.
Ask someone in Tbilisi where they live and you will hear a neighbourhood, not a raion. That is how listings are written too, which is why the pages on this site are organised by neighbourhood and not by the administrative map.
The old centre, roughly Sololaki, Mtatsminda, Vera and across the river Chugureti, is pre-Soviet building stock: high ceilings, courtyards, walkable streets, and often no lift and nowhere to park. Flats there are small for the money and unlike anything newer.
Vake and Saburtalo are the two established districts west of the centre. Vake is greener, has the park and the university, and is consistently the dearer of the two. Saburtalo is denser and newer in parts, and its main advantage is that the metro line runs the length of it.
Didi Dighomi and Gldani in the north are where the square metres are. Blocks are newer or Soviet-era but large, and the same money buys noticeably more floor space than in Vake. The cost is the commute, which is the thing to test at the hour you would actually travel.
East of the river, Isani, Samgori and Varketili are the cheapest per square metre in the city. The building stock is mostly Soviet blocks, and the metro reaches all three, which is the single thing that makes the distance workable.
In practice the decision is made by the metro map and by traffic at the hour you travel, more than by the neighbourhood's reputation. The pages for each neighbourhood on this site show what is actually listed there today and what is being asked for it.
This is a plain description of how the process usually goes, not legal advice. For anything that matters, confirm it at a Public Service Hall or with a lawyer.