The Fourier integral is one way to calculate the Fourier transform of a function:
$$\hat{f}(\omega) = \int_{-\infty}^{\infty} f(x) e^{-i\omega x} dx$$
This definition only makes sense for certain kinds of functions. Since ##|f(x) e^{-i \omega x}| = |f(x)|##, the Fourier integral is defined if and only if ##|f|## is integrable. (In the case of Lebesgue integration, we say that ##f \in L^1##.)

But the Fourier transform can be defined for a larger class of functions, and even some objects that are not functions.

For example, we can define the Fourier transform of a function which is square-integrable but not integrable (i.e. ##f \in L^2 \setminus L^1##). To do this, we approximate ##f## by a sequence of functions ##f_n \in L^1 \cap L^2## and define ##\hat{f} = \lim \hat{f_n}##. Of course there are lot of details to check, such as the existence of such a sequence, and the fact that the limit does not depend on a particular choice for the sequence.

It is also possible to define the Fourier transform of certain types of distributions, which can be thought of as generalized functions. See here for example:

^I agree. The Fourier integral is a method of calculating the Fourier transform. In many cases it is not useful to distinguish between the two. Be aware that there are different Fourier transforms and using a slightly different one can cause confusion.