In the Name of Allaah, The Most Merciful, The Ever Merciful.
The evidence of the obligation of letting the beard to grow is the following Hadeeth:
Ibn ‘Umar, may Allaah be pleased with him, reported that the Prophet, sallallaahu ‘alayhi wa sallam, said, “Do the opposite of what the polytheists do; spare the beards and shave the moustaches.” [Al-Bukhaari]
However, there are other Hadeeths that implied commandments without entailing obligation, such as the Hadeeth in which the Prophet, sallallaahu ‘alayhi wa sallam, said, “Jews and Christians do not dye [their hair and beard], so act differently from them.” [Al- Bukhaari, Muslim and Abu Daawood]
It is known that the righteous predecessors unanimously agreed that dying the hair is not obligatory.
I want to know why the former Hadeeth denotes obligation, while the latter does not, while both of them imply a commandment to act differently from non-Muslims.
May Allaah reward you.

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.

We assume that the honorable questioner has read the scholars’ opinions regarding the ruling on dying the beard in their comments on these Hadeeths; hence, we will not mention them here.

As for the answer to his question, the commandment to let the beard grow came in several forms and using various phrases, which proves that this ruling is stressed. This was also supported by the application of the Prophet, sallallaahu ‘alayhi wa sallam, and his honorable companions, as it was not reported that any of the righteous predecessors shaved his beard.

Furthermore, letting the beard grow is the practice of the Prophets, may Allaah exalt their mention, and one of the characteristics of the sound instinct. Allaah Almighty Reported that Prophet Haaroon (Aaron) said (what means): {O son of my mother, do not seize [me] by my beard or by my head.}[Quran 20:94]

We also assume that you know that if Allaah Almighty or His Messenger, sallallaahu ‘alayhi wa sallam, made a commandment, then it should be understood as denoting obligation unless otherwise proved. (This rule is well-known to the Islamic jurists and scholars of Usool Al-Fiqh.)

Hence, the commandment to let the beard grow is understood as denoting obligation based on this rule and the application of the Prophet, sallallaahu ‘alayhi wa sallam, and his honorable companions, as well as all the righteous predecessors.

The commandments to dye the hair, pray while wearing shoes and the like are not understood as entailing obligation because  many of the honorable companions did not do that in spite of that fact that they knew about these commandments, and because the Prophet, sallallaahu ‘alayhi wa sallam, did not stress them.