What's the difference between gross fixed capital formation and gross private domestic investment?
The term gross private domestic investment (GPDI) seems to be used only by the US. The term gross fixed capital formation (GFCF) is used by the rest of the world (but not by the US). (This is thus simply the US being exceptional as usual by using its own terminology and methodology.)

Roughly, the difference is this: Ignoring inventories, GFCF includes government investment, while GPDI does. More precisely, the difference is this: $$\begin{alignat*}{1}
\text{GFCF} &amp; =\text{Gross private domestic fixed investment}+\text{Gross government fixed investment}\\
 &amp; =\text{GPDI}-\text{Change in private inventories}+\text{Gross government fixed investment}.
\end{alignat*}$$

To elaborate just a little, the rest of the world follows the UN System of National Accounts (SNA, 2008), while the US does not. In the usual identity $Y=C+I+G+NX$, we have the following differences between the UN SNA and the US:





If you are looking at countries other than the US, then you may not be able to find data for GPDI so easily. In which case, you may want to use GFCF.

Conversely, if you're looking at only the US, then you'll probably want to use GPDI.

Of course, in your analysis, you should be clear about what precisely your statistic, be it GFCF or GPDI, is supposed to measure.