If you're interested in using stone for your flooring, travertine is no doubt on your short list. The varieties of colors and designs available make travertine competitive with granite, but the two stones have some important differences.
Travertine is a sedimentary rock. It's often as a building material, both because of its elegant appearance and because of how easy it is to cut. Travertine comes in a wide variety of colors, shades, and styles, which makes it easy to match with any design scheme. It usually contains naturally forming veins that produce interesting and aesthetically pleasing patterns.
Because of the myriad appearances travertine can have, it's comparatively easy to find a new tile that matches your old ones in the unfortunate event that you must replace a tile due to wear or damage.
However, travertine has its drawbacks. It is very vulnerable to damage from acidic liquids. In fact, it can be damaged beyond the point of repair if acid is spilled on it, requiring the damaged tiles to be removed and replaced.
Additionally, travertine is a porous material. This is mitigated somewhat by the fact that when travertine is prepared to be used as a flooring material, the surface holes are filled. However, wear and tear from foot traffic can slowly remove the upper layer, exposing the inner holes and requiring that they be refilled to avoid further damage.
Granite has stood the test of time as a flooring material. It's cost effective, being very cheap in the long term due to the fact that once you buy it, it rarely needs to be replaced. This is because granite is very resistant to all types of damage. Granite resists damage from all kinds of liquids, including acid. Additionally, it's very easy to clean and maintain, usually only requiring infrequent rinsing with water, and occasionally soap. Granite is also very hard, resisting all types of scratches. Granite can be brushed or scrubbed easily if need be without damage resulting.
However, granite is not perfect. Though it is very hard and scratch resistant, it can be scratched by quartz. While avoiding rubbing a particular kind of rock on your floor seems easy enough, the truth is that accidental quartz scratches are quite possible. It is possible for people to accidentally bring quartz sand into a homes on the bottoms of their shoes, where it will be scuffed back and forth across a granite floor, causing scratches. Additionally, granite does require a large initial investment, even if it does pay off in the long run. Lastly, it comes in relatively few colors.