There are two questions here: which version of the theory is easiest to get off the ground axiomatically, and which version is more convenient to work with in applications?
For the first question, it's very much a matter of taste. Adic spaces are genuinely topological spaces, whereas Tate's G-topological spaces aren't; but there are many more, and weirder, points in them whose geometric significance takes some getting used to.
At risk of treading on some toes, I'm going to point out that Huber's adic spaces were available in the literature for at least 20 years before they started to become really popular. If there were a decisive advantage to setting up the theory in terms of adic spectra rather than G-topologies, then number theorists would have abandoned Tate's theory en masse in 1995 or so; and they didn't. The benefits offered by Huber's foundations weren't persuasive enough to to outweigh the "first-mover advantage" given by 30 years' worth of literature written in Tate's language. Adic spaces caught on because Scholze showed they could be used to do radically new things that were impossible in Tate's rigid geometry -- not because they allowed you to re-prove or re-visualize existing theorems in nicer ways.
As for the second question, I think the correct answer is "both". There are some (mostly younger) mathematicians who, whenever rigid spaces are mentioned, smile indulgently at the folly of their elders and assume that anything written in this language is obsolete or misguided, a bit like teenagers laughing at their parents' CD collection. This is a misconception, since rigid spaces over a nonarchimedean field K can be identified with a subcategory of adic spaces over K, and a rigid space and its corresponding adic space have the same sheaf theory (equivalent as topoi). Hence, when you want to apply p-adic analytic geometry to actually do something, it very often doesn't matter whether you write $Max(A)$ or $Spa(A, A^+)$ -- their underlying sets are hugely different, but that's usually not relevant if you're writing a research paper as opposed to a textbook. Indeed, in recent literature it seems to be quite common to simply redefine "rigid space over K" to mean an adic space which is locally of finite type over K. So the large corpus of work written in Tate's language remains useful, and a working number theorist nowadays who isn't familiar with the older language is at risk of needlessly reinventing the wheel. [I'm sure Wojowu already knows everything I've written in this paragraph, but I'm putting it in for the benefit of other readers of this question.]