For example: Is there a significant difference between a 15.75 mm crochet hook and a 16 mm crochet hook?
I have a pattern that calls for a 16 mm crochet hook, but all I was able to find was a 15.75 mm crochet hook.
You'll probably be fine. One quarter of a millimeter isn't a huge difference, particularly considering how large the hook is.
The reality is that the hook size is just a suggestion. Different crocheters will need to use different hook sizes depending on their tension. Someone who crochets very loose will need to use a smaller hook for the same project while someone who crochets tight will need to use a larger one.
Assessing gauge is necessary when starting many projects. Some are pickier than others. A shawl/blanket/scarf is unlikely to need you to find your gauge first as you're not working to size anything. Hook size and yarn weight is much more flexible in these projects. On the other hand, clothing (like a sweater) will require you to use the proper gauge for you to get the item sized properly.
If gauge matters for this project, work a test swatch using your hook and compare it to the recommended size. If your pattern says that 8 stitches is an inch and your test swatch gets only 7 stitches to the inch, you need a smaller hook. If, however, you get 9 stitches, you'll need a bigger hook.
In case of small hooks, especially the tiny steel ones, it makes a difference. Look at at a picture (from this post) of two hearts crocheted using the same pattern, but different hooks:
The difference of a quarter of a milimeter is the same as with your hook and pattern, but for tiny hooks this is in fact a difference of 3 sizes (increments are 0.1mm). The larger heart is twice as large as the smaller one.
Additionally, if your gauge is off using your hook, but you like the fabric it's giving you, remember that you might need a different amount of yarn for your project.
I have noticed that it makes a difference. I like to look for measurements in crochet projects or come up with my own measurements to use as guidelines so if I use a smaller or larger hook size, I can decrease or increase my stitches to reach the desired measurement. That being said, you may still notice a difference in the tightness of your work depending on your project (what stitches you are using), type of yarn, tension, etc.