The snowsong is stationary
Were the snowsong spell effect to move with the caster, the spell would likely have header entries like those of the 6th-level Sor/Wiz spell antimagic field [conj] (PH 200) and be, instead, an emanation: "An emanation spell functions like a burst spell, except that the effect continues to radiate from the point of origin for the duration of the spell" (PH 175). The snowsong spell effect's not an emanation and has no special rules saying otherwise, so its effect stays where it was put like most other non-emanation effects that don't specify otherwise.
In other words, the 6th-level bard spell snowsong [evoc] (Frostburn 105) creates an immobile area. Although the spell's area is measured from the caster, like the similar 1st-level Sor/Wiz obscuring mist [conj] (Player's Handbook 258) and the extremely similar 6th-level bard spell dirge [evoc] (Spell Compendium 65), the snowsong spell effect is stationary.
Just to be clear, the snowsong spell's description begins with the following:
When you cast the spell, you must designate all creatures in the area as either allies or enemies. (Fr 105)
And it ends with this:
These benefits remain in place as long as the spell persists and as long as the target remains in the spell’s area. If a creature leaves the spell’s area, all effects end for that creature until it returns to the snowsong’s area. (ibid.)
(Emphases mine.) Needless to say, casters usually designate themselves as allies and tend to want to stay in that area. (Its immobility is also likely why this guide lists the snowsong spell as only an average spell choice for the typical bard.)
Note: While the mist spell's description says that its effect "is stationary once created," this reader assumes that this reminder is because of the spell's status as a 1st-level spell, its clarification right there so players new to the spell needn't look up these rules. Because of its Range entry a DM may rule that a caster can affect spells like obscuring mist and snowsong with the Complete Arcane feat Persistent Spell (81) (see, for example, here; in such cases, effects would remain for 24 hours right there where they were cast, totally immobile.