What happens when the amplitudes of interfering waves is different in the phenomenon of beats?
As an illustration to the other answers, here are two waves that we'll later superimpose (their frequencies ratio $\nu_1/\nu_2=15/16$):
Their superposition with identical amplitudes of 1 will look like this (red is the envelope, green – the sum):
And here's what happens when one of the waves has amplitude 1, while the other 0.7:
Beat frequency is as you say when the difference in frequency is low enough for us to make out a beat. When the beats occur the signal is ideally interfered into $0\%$ amplitude. If the amplitudes of the two signals however are very different we'd have a reduction in intensity but not an attenuation to $0\%$ but maybe instead to $90\%$ if one of them is $10$X the other one. This would not be as easy for us to detect.
In other words, the amplitude does not need to be identical, but it helps us to show the phenomenon.
It's not necessary for the interfering waves to have the same amplitudes. When the interfering waves have slightly different frequencies and different amplitudes, the resultant wave also shows the phenomenon of beats. But unlike the case when both waves have the same amplitude, there will be no time when the amplitude of the resultant wave is zero. Or in other words, there will be no moment of silence which you'd normally observe when amplitudes of interfering waves are same.
If you're interested in this topic, I'd highly recommend you to do the experiment yourselves using an "Online Tone Generator". I used this one for the experiment.
Open the above website in two different tabs. The default frequency will be at $440$ hertz and the default volume will be at $75$%. Leave the first tab aside, and in the second tab, change the frequency slightly, say to $444$ hertz. Now, press the play button in both the tabs separately. Now you'd observe the phenomenon of beats with a beat frequency of $|444-440|=4$ hertz. This is the ordinary case when both the interfering waves have the same amplitude but slightly different frequency.
Now in order to experiment the case in the question i.e., amplitudes of the interfering waves are different, you must change the volume in either one of the tabs to a lower (maybe a higher!) value say $15$%. Now press the play button in both the tabs. Again you'll here the phenomenon of beats, but this time there will be no moment of silence. The minimum amount of sound you'd here will be some non zero value.
I also experimented it (using the values mentioned above) before I posted this answer. It's really interesting to experiment something you learn from your textbooks.
Note: If you find any difficulties in using the tone generator to observe the phenomenon of beats, watch this video from Bozeman science. Although he does this experiment for component waves with same amplitude (or less specifically, volume)