We have used variable-temperature scanning tunneling microscopy to study the aggregation of two-dimensional Ag clusters on Pt(111). A transition from randomly ramified to dendritic fractal growth is observed in the diffusion-limited regime. Atomic-scale observations have identified the anisotropy of edge diffusion as microscopic origin of this crossover. Dependent on the deposition flux, this anisotropy is either amplified to the macroscopic-cluster shape and trigonal dendrites result, or it is degenerated and randomly ramified fractals occur. Our study elucidates the close relation between fractal and dendritic pattern formation in diffusion-limited aggregation on a two-dimensional lattice.