Would a Displayport-Thunderbolt cable unlock better performance from my external display than an HDMI 2.0 cable?
For this connection you could have four bottlenecks:

Input connector on your monitor
Output connector on your Mac/PC
faulty cable/lower-spec cable
GPU (graphics processing unit) limitation



As you state, the input connector on your monitor is an HDMI 2.0
port, which is capable of 4k@60fps.
A mid 2014 13&quot; MBP does only support 3840x2160@30fps output over the HDMI port. This may be at least one of the reasons you are not getting the higher refresh rate.
You can simply test the cable with another device. I also would not really suspect it, unless you have a reason to believe it is physically damaged.
The MBP specs also state that the output is too limited for your high refresh rate monitor. There has also been a similar question on AD:


Simultaneously supports full native resolution on the built-in display and up to 2560 by 1600 pixels on up to two external displays, both at millions of colors

What this means for you
You will not be able to get a higher refresh rate, since you are using an older HDMI port and the MBPs video output is too limited. So even if you would get a Mini-DP → HDMI 2.x cable, you shouldn't  be able to get a higher refresh rate.

Note: On your machine the Thunderbolt 2 connector can act as a Mini-DP connector, which is capable of higher resolution/refresh rates than some older HDMI ports. Yet, this depends on the exact comparison. Have a look at the Wikipedia page about the different HDMI specifications and Display Port connections.
Double check which MacBooks support what monitor resolutions and refresh rates on Everymac.