Will a 4K external monitor create a sharper display when connected to my MacBook Pro
The "sharpness" of a display is defined by its pixel density and measured in ppi (pixels per inch). So it is the combination of size and resolution that makes a display appear sharp.
Unless the 4K monitor is extremely large the pixel density will be higher then on your 1920 x 1080 pixel display.

You can check compatibility on Using 4K displays, 5K displays, and Ultra HD TVs with your Mac.

In your case, your Late 2013 MacBook Pro 15" supports 4K displays. However you also have to use a cable that supports the resolution accordingly. For example:

HDMI:


3840 x 2160 at 30Hz
4096 x 2160 at 24Hz