The head of the new part serves to accentuate and repeats the same pitch a few times. The new part serves mainly to give the piece a new rhythmic pulse. With regard to this piece, the parts that are entering at 0:15 do not really seem to carry a new or separate melody. So in those cases, the extra melody is not truly independent. Or although a melody might be playing different pitches, it might do so in an identical rhythm.
Although the melodies might have a different rhythm, a melody might have the effect of emphasizing only particular accents of a main melody.
'Independence' is not a matter of all or nothing. The term is about a technique or texture, and as such it may be applied to a section, or a movement or even an entire composition. Both rhythm, pitch but also timbre can be used to achieve independence. The term counterpoint is used when the melodies that are sounding simultaneously are independent and are more or less of equal importance. As pointed out, the general concept of two (or more!) melodies sounding together is counterpoint ('note against note').
