The general historical pattern with these kinds of Germanic surnames, most exemplified by German and to a lesser extent English, is that an adjectival form is subject to nominalisation, after which the article (and some of the inflection) tends to get lost.

Example: A man nicknamed brown John, becomes John the Brown, which ends up as John Brown.

To show the loss of inflection, I'll use the modern German equivalent as well: brauner Johann > Johann der Braune > Johann Braun.

(Note: the example isn't exactly accurate, as for most of the medieval period (and in some way up until today) the brown John-form very much coexisted with the the Brown variant; the latter is definitely a later development.)

In this regard, Dutch is being more conservative, generally preserving the surname practices of the (late) medieval period and is atypical this respect. So to answer your question: Dutch retained them, whereas English and German lost them.

German lost the practice of using either the adjectival form or the use of a definite article during the medieval period. In the Middle Ages, to use your example, Peter Weiß existed alongside forms such as Peter der Weiße or der weiß(e) Peter. In fact, in Middle High German texts, the form with a definite article appears to be more common, especially in the North and Central areas of Germany.

In Old English, most of these kind of adjectival surnames (such as White) were inflected, though they can appear otherwise to the modern eye. This is because Old English word order was much more free compared to modern English, so Peter hwita should be translated as white Peter even though using the same word order, it would spell Peter White.

The use of definite articles was rare in Old English and is often considered typical for the middle stages of the Germanic languages. Middle High German, Middle Dutch and Middle English all use articles and prepositions to a far greater degree than their ancestral forms.

The transition from Old tot Middle English, was also far more abrupt (due to the external linguistic influences accompanying the Norman Invasion) compared to its continental counterparts, which might account for the relative lack of an intermediary stage. This is somewhat counter intuitive though, as prepositions and articles were quite common among the medieval French names that the Normans brought over, though even these surnames began to be dropped in English during the 14th century. There is no commonly accepted explanation as to why this happened. Scribal errors or a change from nicknames to hereditary surnames has been suggested, but these aren't considered to be very convincing. A French influence (French typically putting the noun in front of the adjective; blue cheese > fromage bleu) has been suggested, but isn't widely supported; and wouldn't explain the same process in German; which had a much more limited French influence during its development.

The most likely explanation is that it was caused by convenience; somewhat oversimplified: a person would be in a conversation and refer to a man name John. The other person would ask to specify (Which John?) to which the reply would have been (the) White.

Of course, this explanation still doesn't account for the retention of articles in Dutch, which has been suggested to lie in the already prolific use of various prefixes (caused by high population density and early urbanisation throughout the Low Countries) which reinforced the practice of using articles; with which the prefixes are often combined.