Is it possible to compute the signal-to-noise ratio from these 2 audio clips alone?
Maybe.
In order to calculate SNR, the signals must have the same calibration or scaling. Verifying or calculating the scaling will require information on how the recordings where done: microphone type and sensitivity, pre-amp gain, A/D gain, any type of scaling or normalization done by the recording software, etc. That tends be very tricky so in most cases you will need the recordings to be either done with the exact same setup or contain an actual calibration signal.
Next you need to determine how "clean" the speech recording is. Is the "no noise", "same noise" or "different noise" from the noise-only recording. You can calculate SNR for "no noise" (which is easy), from "same noise" (if it's in a suitable range) but not for "different noise". "Same" doesn't require the exact same recording but the same general noise type (HVAC, traffic, background music, concurrent talker, etc) and the same level.
Finally you need to decide on how to handle frequency and time. Audio SNR often varies with time and varies A LOT with frequency. If your noise and speech are reasonable stationary, you can process the entire recording for a single SNR. If the noise varies a lot, you may have to chop the recordings up into smaller stationary sections and you need end up with SNR as a function of time. That depends a bit on your specific application and what exactly you are planning to do with the SNR.
Frequency dependencies is typically handled by applying a frequency weighting filter that models what frequencies are most relevant to human auditory perception. The most common is the so-called "A weighting". See for example https://en.wikipedia.org/wiki/A-weighting
In conclusions: if your signals are scaled identically, if the noise is stationary, AND if the speech signal is noise free, you can calculate the SNR simply as
$$SNR = 10 \log_{10}\frac{\sum s_A^2[k]^2}{\sum n_A^2[k]}$$
where $s_A[k]$ is your A-weighted noise free speech signal and $n_A[k]$ your A-weighted noise signal.