Given any data set $(t_i,y_i)$, to say that we interpolate those data points means to find a function $f(t)$ so that $f(t_i) = y_i$. We know several functions $f$ for example the Monomial basis, Lagrange, and newton basis. For example, using the newton basis, our $f$ is given by
$$ p_{n-1}(t) = x_1 + x_2 (t-t_1) + x_3 (t-t_1)(t-t_2)+... + x_n (t-t_1)(t-t_2)...(t-t_{n-1}) $$
where $x_i$ are to be determined from the system $p_{n-1}(t_i) = y_i$.
Now, we can use this to write functions are polynomials, correct? For example, take $f(t) = e^t$. For simplicity, lets take two data points $(0,1), (1,e)$. we obtain $p_1(t) = 1 + (e-1)t$ as the interpolant of $e^t$.
If we do the comparison with Taylor's polynomial. We don't think of it as an interpolant, but as an approximation. What is the main difference between an interpolant and the Taylor's approximation?
There are several differences in spirit and in terms of results between interpolating a function with a polynomial and approximating it with a Taylor expansion.
The main difference is that a Taylor expansion approximates a function around a point. The polynomial will be exact on the point and close to the function in a neighborhood of the point. Elsewhere it could be arbitrarily far from the approximated function. An interpolation on $n$ points (with a polynomial of degree $n-1$) will be exact on all the interpolation points, and (if Runge's phenomenon is not taken into account) the interpolation polynomial will get closer to the approximated function as the number of points and the degree of the polynomial grow. Moreover, interpolation does not require the knowledge of the derivatives of the function, which is essential for a Taylor expansion.
To sum it up: if you want to approximate a function around a given point, and you have access to derivatives of the function, then you may want to employ a Taylor expansion. If you want to approximate a function on an interval where you can access some function values (but not derivatives), you could employ an interpolation polynomial.