This is a bit tricky. Both Proteus vulgaris and Pseudomonas aeruginosa are Gram-negative and rod-shaped. According to Bergey's Manual of Systematic Bacteriology, Proteus vulgaris tests positive for sulfur reduction, urease production, tryptophan deaminase production, and indole production. Pseudomonas aeruginosa reduces sulfur and nitrate, and has a characteristic green color. The API 20E Identification System is a test strip that can be used to identify Proteus vulgaris versus other bacteria.
An even easier way to test this is the gelatinase test.
Or, slant the bacteria on a Triple Sugar Iron slant or place them in a SIM agar deep. P. vulgaris tests positive for Hydrogen Sulfide production, changing the media to a black color around the organism, while P. aeruginosa does not produce Hydrogen Sulfide.