Yes, No, Maybe, but it's not as simple as you imply.
I recently acquired a 32 inch 4K monitor and it's nice for the price ($400). It doesn't support high frame rates required for gaming but it handles photos nicely.
The 16 million colors thing comes from 8-bits each of Red/Green/Blue for a total of 24-bits, or 2^24 which is 16 million (more or less). The billion colors thing comes from 10-bits per each R/G/B. You would need a graphics card capable of 10-bit color. You would need to be working in RAW. You would need a monitor capable of 10-bit color. The first two requirements are fairly straight forward, but a 10-bit monitor is going to be really really expensive.
Monitors vary tremendously in contrast, gamma, color balance, and the ability to adjust them.
The standard response to questions like this is to suggest calibrating your monitor. This is certainly a good thing to do in order to make the most of your monitor, but a poor monitor is still a poor monitor. Calibration allows you to generate an ICC profile for you monitor. In lieu of direct calibration, using a published ICC for you monitor is a good starting point.
So should you upgrade?
Ultimately that's up to you, a new monitor won't be magical but your current monitor leaves much to be desired. A big 4K monitor is much nicer to work with, but be aware it takes substantially more video horsepower to run.