Both Floyd-Warshall and Bellman-Ford handle negative edges. Bellman-Ford finds shortest paths from a single source in time. Floyd-Warshall finds shortest paths from all sources in time. Use Bellman-Ford if you need paths from one source.
Use Floyd-Warshall if you need all-pairs distances or have many queries. Bellman-Ford is better for sparse graphs with one query. Floyd-Warshall is better for many queries.
Space complexity is for the data structures used.