Change in feel, feedback, and stroke
As strings lose tension, you may feel that the racket is "going dead," "getting mushy" or "losing its punch."
Obviously it is not, since the ball is going faster and farther. But what is happening is you have lost the crisp feel you have become accustomed to. Crisp means more shock, but shock is feel. The only sensations of striking a ball that your hand feels are shock and vibration. This is your feedback mechanism. When the feel is the same every time, your response is to groove the stroke; when it is different, you respond by continually adapting and adjusting your stroke.
The other feedback that changes is auditory. The sound the strings make changes.
As tension goes down, hitting the ball goes from a "ping" to a "thud." Players may interpret these sounds differently as to what they mean about the cleanness of their hits. When this sound changes, so does the player's psychology. It affects what they think they are doing, how they are performing, what the results are and whether they are in the zone or not. All this affects the mental and physical approach to the next shot.
Change in spin and stroke
It has been shown that string tension has very little impact on spin. A ball fired obliquely at the same racket with different tensions rebounds at about the same spin. However, if tension goes down and you are thus hitting the ball deeper (too deep), your natural response will be to either hit it more softly or to add more spin. The loose strings don't cause more spin; rather, they cause you to add more spin. Again, these are usually unconscious adjustments to your strokes as your day-to-day racket performance changes. Your strings change your strokes daily. And you thought you were just having a bad day.
More string movement, less durability, less performance
Strings slide across each other more easily at lower tensions. The main strings move back and forth over the crosses. This has a couple of obvious effects. First, it shortens the life of the strings as they saw through each other. Second, if you don't move the strings back into place after each hit, you will have an uneven string pattern and will end up with an uneven power and control response across the string face. This will affect the bounce of the ball and you will be making stroke adjustments to compensate.
Consistency is key
If you don't restring often enough, chances are you'll spend much of your tennis life compensating for your changing string tension instead of honing your swing. Consistency is the key.
You want to play with the same racket as you did yesterday--one that will behave the same way in the same situations so you can let your muscle memory take care of swinging while you figure out what you've got to do to beat the guy on the other side of the net.
So how often should you restring? If you're a frequent player, the answer is most probably, "More often than you do now!" For most of the recreational playing population, doubling their restringing frequency from what it is now would not be overkill.
More objectively, a good guide is to restring when the stringbed stiffness (measured by equipment in a pro shop) has dropped by 20 percent from the reading immediately after stringing. You'll need to experiment to find what works and feels best to you.
In the end, though, you will find that this is definitely a case where more is better.