Mathematicians model ideas and study their properties. In mathematics, a vector space models a set of things that behave like like little arrows when added together and multiplied by numbers.
The mathematical definition of a vector space is a set with an addition operation and a multiplication by number operation that follows 8 rules. Anything that meets the definition is of interest to mathematicians. For example the set of continuous functions on the interval [0,1] is a vector space.
Mathematicians discover things like every vector space has at least one basis. All bases of a given vector space have the same number of vectors. This number is called the dimension.
Since a tensor space meets the definition of a vector space, it is a vector space.
Physicists often find that mathematicians invent useful tools. But physicists are interested in modeling the behavior of the universe. They use the tools in different ways than mathematicians do. For example, they are often less interested in mathematical rigor.
For the most part, physicists use vectors to model things like space or momentum. They find vectors are useful if they have a norm (or metric or length) and all the components are the same kind of quantity. The forward direction is space. Sideways is space.
If all the components are the same, you can change the basis and still use the vector space to model the universe. If I am facing forward and you are at 45 degrees to the right, we both can use $F = ma$.
Because of this physicists are very interested in how vectors transform when you change the basis.
And this is where tensors differ. Tensors of rank 2 transform differently than tensors of rank 1. So to physicists they are different objects.
As an aside, note that in relativity, one of the components is different than the others. Physicists have defined a useful, not-quite-kosher metric. They have found that when you change the basis, you get a useful model of the universe from the point of view of an observer at a different velocity.
Another aside, mathematicians would consider the phase space of statistical mechanics to be a vector space. But physicists don't much care about the vector properties of phase space. They don't add vectors or change the basis. They just follow the trajectory of a point as a system evolves.