What is the difference between a lock-in amplifier and a superheterodyne receiver?
A lock-in amplifier is really just a type of homodyne receiver. I think a better approach would be to do the comparison between homodyne vs super-heterodyne, which makes the comparison more general. Actually, comparing homodyne vs heterodyne would be more general but to keep the answer short lets go with superhet.
Homodyne Receiver
A homodyne receiver's goal is to transmit a reference signal which is then mixed with the received signal to retrieve some type of phase or frequency information. Depending on the specific system, the mixing and detection process yields a near-zero or exactly at-zero signal that can then be processed further. Digital processing lends itself better to handle signals centered at zero. All of this is good: signals at or near baseband have decreased processing demands.
Homodyne receivers are much simpler than their heterodyne counterparts but are not as flexible. You are stuck comparing your reference signal to the received one. Adding more waveforms will make the receiver more complex and you'll reach a point where it is no longer worth to use this method.
In terms of detection performance, the homodyning process itself is not somehow better than other methods. If I choose a homodyne receiver over a heterodyne it will be because the application I am designing for can be accomplished without spending the extra money on a heterodyne.
With a given set of hardware (mainly antenna and average power on target capability) and assuming ideal signal processing conditions, once your signal bandwidth is chosen, your raw performance in terms of SNR and detection capabilities is sealed. This is assuming that you are using a matched filter, which is usually the case.
To summarize, a homodyne receiver:

Mixes a transmitted refrence signal to the received signal to immediately yield a  zero or near-zero signal.
Overall receiver is simpler and has decreased bandwidth requirements so circuits and sampling can be done cheaply.
Not as flexible as other receivers. The transmitted waveform is the reference you must use.
Isolation between receive and transmit channels must be very good since many systems of this type usually do simultaneous transmit and receive.

Super-Heterodyne Receiver
Similar to a homodyne, a superhet converts from an RF frequency to a lower one. Except this time, instead of going to zero or near-zero, we go to some intermediate frequency (IF) that can be tens or hundreds of MHz. Here we don't need our transmitted signal to be a reference. We only need a local oscillator to shift the RF to an IF.
By not going straight to a near-zero signal, you have increased flexibility in what you can do. An example of this is waveform agility. You can very easily change the waveform you transmit (assuming the same bandwidth), and the mixing process will always place it at the IF of interest. We cannot easily do this with a homodyne receiver as it is dependent on the reference signal.
As before with the homodyne, the lower IF frequency decreases the sampling and processing requirements of our receiver. However, the superhet receiver is more complex since more filters and mixers need to be introduced for image rejection and other signal conditioning tasks.
Once we're at IF, we can process the signal in the analog domain or perform sampling to further process it digitally. Again, usually some type of matched filter is used.
To summarize, a super-heterodyne receiver:

Mixes an arbitrary signal with a local oscillator to move the signal from RF to a much lower IF.
More complex design due to increased number of filters, mixers, and other components.
Much more flexible. Lends itself better to the use of arbitrary waveforms and tuning within the system's allowed bandwidth.

At the end of the day, the two receiver designs trade between flexibility and simplicity. Depending on the use case, one type will lend itself better than the other.