Nitrogen cycle semantics; nitrification/nitrogen fixation
I went to do a bit of research on this, and think that you can get a satisfactory answer just from the relevant wikipedia articles. Here are some select quotes:

Nitrogen fixation is a process by which nitrogen in the Earth's
atmosphere is converted into ammonia (NH3) or other molecules
available to living organisms.

Nitrogen fixation has the chemical process:

N$_2$ + 8 H$^+$ + 8 e$^−$ $\rightarrow$ 2 NH$_3$ + H$_2$

and is performed by Diazotroph bacteria or archaea. Some of these are symbiotic with plant roots, especially of the family Fabaceae.

Nitrification is the biological oxidation of ammonia or ammonium to
nitrite followed by the oxidation of the nitrite to nitrate.

An example of a nitrification chemical reaction is:

2NH$_4^+$ +  3O$_2$ $\rightarrow$ 2NO$_2^−$ + 2 H$_2$O + 4 H$^+$
2NO$_2^−$ + O$_2$ $\rightarrow$ 2 NO$_3^−$

There are a wide variety of bacteria that peform these steps, and some only do one step or the other. There are also other pathways for reducing ammonia instead of ammonium, for example.
In conclusion, these are two different biologically driven steps in the nitrogen cycle, driven by two different sets of microorganims. There other biologically driven processes as well, such as ammonification, where organic nitrogen compounds in formerly living tissue (or waste) is broken down into ammonia or ammonium.