First, because Anglo-Saxon poets tended not to use rhyme--particularly end rhyme--in their poems, which is a technique poets use to connect lines together in order to help the poet recite a poem to the audience (keep in mind that Anglo-Saxon poetry began as an oral tradition, not written), Anglo-Saxon poets employed initial rhyme, also known as alliteration. Second, the primary effect of the repetition of initial sounds is to "move" the speaker and listener through a line of poetry quickly--think of it as a wave moving from left to right--partly to enhance the listener's ability to memorize a poem. Third, because of its usefulness, alliteration became over time an expected feature of Anglo-Saxon poetry. More specifically, each pair of half lines had to be connected by alliteration. For example, in the first few lines of Beowulf, we have
Beowulf was well known wide-spread his fame--
the son of Scyld in Scandinavia (ll. 18-19)
In this example, in the first two half lines ( each called a hemistich), we have the initial w sound four times in seven words, and in the second, we have initial s sounds in three words in a six word line. The alliteration, one can argue, encourages forward movement for both the speaker and the listener because it creates rhythm (without rhyme).
Given the fact that alliteration is an expected component of all Anglo-Saxon poetry--and that it is a technique to create rhythm without rhyme--it is difficult to argue that alliteration itself is used to differentiate characters. Certainly, alliteration enhances the description of characters, their speech, and their actions, but alliteration enhances description (of place, for example) and narrative in exactly the same way.
In sum, then, alliteration has important effects in Anglo-Saxon poetry in general, but its effects--particularly its creation of rhythm--are not specific to individual characters in the poem.