Can you objectively tell the quality?
Generally, no, there's no accurate way to objectively measure audio quality with enough precision. Simply said: Without asking a set of people for their specific rating, you can never quantify "how much higher" in quality one piece of audio is, compared to another. Quality itself is never absolute—it's highly subjective.
What researchers usually do to alleviate that problem is the following: They define a scale according to which they let a set of people rate the quality, such as a five point one (Bad, Poor, Fair, Good, Excellent). Or you could simply compare certain codecs (but that wouldn't give you very meaningful data). Researchers then use these ratings to model a computer metric that tries to predict the human response.
Objective quality metrics
A rather old but popular objective audio quality metric is Perceptual Evaluation of Audio Quality (PEAQ), described in ITU-R BS.1387 (1998). Basically, it tries to emulate the human ear and generate opinion scores from 1–5, like a human listener would do. PEMO-Q from 2006 is a more recent metric that claims to outperform PEAQ.
These metrics are often tested on single sounds or speech, and not necessarily on music, as it is much harder to estimate the perceived quality from complex sounds, where masking effects are much more prevalent. This metric from 2008 is targeted more towards music.
It seems to me that research in audio quality metrics is not as active as for video quality metrics (a topic I'm working on). You will have a hard time finding a standalone tool that you can use on Windows where you can just plug in two sources of audio and get a result. Most metrics are developed internally at research institutes and rarely see the light of the public. Or they're commercial tools for broadcasters and telecommunications providers.
Here's a MATLAB implementation of PEAQ, but that's all I know of from the top of my head.
What you can do when comparing files
Looking at the problem from a technical perspective, it is important to mention that you can – in theory – guess which file delivers the better quality by just looking at the data rate. This, however, only works reliably when the files contain the same source content, and they were encoded with the same encoder.
That's the whole idea behind the codec listening tests: You take the same source, encode it to different data rates (or quality levels) with different encoders, and then have users compare them. With this "ground truth" data, you could for example say that an MP3 file encoded with LAME at 128 kBit/s sounds equally good as an AAC file encoded with FAAC at 64 kBit/s (these are made up numbers, but you get the idea).
It's even easier if you know that the same encoder was used for both files, because then chances are very good the data rate (file size) is enough to make an educated guess about the resulting quality.
But then again, this only works when the sources are the same. Some codecs perform better for different kinds of music or speech. Some work better at lower bit rates and don't necessarily provide any better quality at higher rates. There are just too many factors to consider.