Why is the specific notation used for term symbols useful?

Apparently it's a historical quirk. Characterizing spectral lines as principle, sharp, or diffuse dates back to the 1870s with the works of George Liveing and Sir James Dewar. Living and Dewar also noted that these lines appear in series. Arno Bergmann discovered a fourth series in 1907, which he labeled as the fundamental series.

If Arnold Somerfeld had had his way in 1919, the f (fundamental) series would have been named the b (bergmann) series in honor of Arno Bergmann. Eight years after Somerfeld published his monograph, Friedrich Hund, who was working on reconciling those pre-quantum spectroscopic observations with quantum mechanics theory published the monograph Linienspektren und periodisches System der Elemente that reverted the names to sharp, principle, diffuse, and fundamental.

That labeling is what you're stuck with now.

Reference:

Jensen, "The Origin of the s, p, d, f Orbital Labels," Journal of Chemical Education 84, no. 5 (2007): 757

I am not sure if I can give you the reasoning for this choice or the full picture when it was developed but from what I take from this is the following.

The focus of the notation of term symbol is not on the description of the electron configuration but rather on the strength of lines and allowed transition between electron levels.

Here are my thoughts. First, some important points from the wikipedia article. With Russel Saunders Coupling there are $(2S+1)(2L+1)$ allowed micro states, with strong coupling between $l$ and $s$ themselves but weak coupling between $s$ and $l$. That means that terms with different $L$ or different $S$ have very different energies but terms resulting from same $L$ and $S$ have very similar energies, those form a multiplet term with $2S+1$ when $L >S$ and $2L+1$ when $L<S$ components and occur as such in the spectrum. Hence, the prefixed exponent in the term tells you this directly.

So instead of the value of the resultant spin the notion of the term symbol emphasizes the multiplicity of the term. The reason for this comes from the emission intensity $I$ of a spectral line: $I\propto A_{ki}$, where $A_{ki}$ is the transition probability between state final state $k$ and initial state $i$, which itself for a electric dipole transition is $$A_{ki}\propto\frac{g_i}{g_k}$$, the multiplicity of those states.

So, given two terms, after checking selection rules (for which you need $\Delta S$, $\Delta L$ and $\Delta J$) you can have a rough idea how difficult it is to measure the transition line, and hence focus is not on the description of the electron configuration.

Some useful sources:

• Herzberg, Molecular spectra and molecular structure. I. Spectra of diatomic molecules
• NIST atomic spectroscopy