  Two commonly studied compactifications of Teichm\"uller spaces of finite type
surfaces with respect to the Teichm\"uller metric are the horofunction and
visual compactifications. We show that these two compactifications are related,
by proving that the horofunction compactification is finer than the visual
compactification. This allows us to use the simplicity of the visual
compactification to obtain topological properties of the horofunction
compactification. Among other things, we show that the horoboundary of
Teichm\"uller space is path connected and that its Busemann points are not
dense, we determine for which surfaces the horofunction compactification is
isomorphic to the visual one, and we show that some horocycles diverge in the
visual compactification based at some point. As an ingredient in one of the
proofs we show that extremal length is not $C^{2}$ along some paths that are
smooth with respect to the piecewise linear structure on measured foliations.
