What is the difference between 1N4001 and 1N4007 other than their maximum reverse voltage?
The answers by @Vasiliy and @johnfound are incorrect. 1N400x diodes are not all identical except for "accidental" manufacturing variations.

Diodes with higher reverse voltage ratings are intentionally manufactured with lighter doping so that the depletion region for a given reverse voltage is wider than it otherwise would be. The disadvantage with lighter doping is that the forward resistance and voltage drop for a high-voltage diode is higher than it would be for a lower-voltage diode.

So, you can use a 1N4007 for all of your applications, but your circuit efficiency will be slightly higher if you use a more appropriately-rated diode in low-voltage applications.