Diffraction pattern vs Interference pattern

The centre of the bright fringes that you see using a diffraction grating are in fact in exactly the same position as those produced by two slits with the same separation as that between adjacent slits when using a diffraction grating.

Given that the grating equation for the n$^{rm th}$ maximum is usually written as $nlambda = d sin theta_{rm n}$ and it the same for the double slit you can say that the fringes are not equally spaced.

However for the normal double slit arrangement the angle $theta_{rm n}$ is small and so the approximation $sin theta_{rm n} approx theta_{rm n}$ can be used.
So $y_{rm n} approx D ,theta_{rm n} = frac{n lambda,D}{d} Rightarrow y_{rm n+1} -y_{rm n} = Delta y = frac{(n+1) lambda,D}{d} - frac{n,lambda,D}{d} = frac{lambda,D}{d}$

This results in fringes which are observed to be equally spaced.

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The advantage of using a diffraction grating is that the bright fringes are narrow and much brighter than those from a two slit arrangement as explained here.

The width of a slits controls the diffraction envelope ie modulate the intensity of the interference fringes.

Just getting into definitions here, interference refers to the action of waves meeting with each other and combining constructively or destructively. A diffraction pattern on the other hand is defined primarily by interference but also by the source's interaction with an edge or slit. A diffraction grating can create a diffraction pattern, whereas a Michelson Interferometer does not. A diffraction pattern typically has well defined orders of light, but interference patterns (that have not been created by diffraction) are usually fuzzier, although that may not always be the case.