Do short keycaps make any difference?

[bluejeans]

My first mechanical board is an actual cherry branded one with blue switches.  I didn't know much about mech boards at the time and it seems it's for typists, the keycaps aren't very tall compared to gaming keyboards.  Does it make any real difference?  Would it be possible to simply use higher keys from another board?  The key travel doesn't feel what it's meant to be, compared to blacks and greens.

[Tactile]

I think you're referring to what is called the "profile" of the caps. There are many. Your Cherry board would have Cherry profile caps. A set of caps from Tai Hao would be taller and wouldn't cost a whole lot.

[Findecanor]

Key travel is the same.
If you use one of Cherry's regular G80-series keyboards in grey or black then inclination of the surface on the keycaps and the size of the surface area is also practically the same.
It is just the overall height of the keyboard that is different.

The "MX-Board 2.0" and "MX-Board 3.0" have keys in a profile differing than Cherry's norm - these are flatter but they are just as high as the "home row" of Cherry's regular keys. The keys have skirts and the border around the keys is higher to give the illusion of low-profile keys.

Most "gaming" keyboards have keys in the so called "OEM-profile". You can get replacement key sets in that profile from Max Keyboard and WASD Keyboards and there have also been sets by Ducky.
Cherry's Caps Lock and Space Bar keys tend to have the mounting stems somewhat offset to that norm though so a kit in OEM profile might not fit your board.

Most key sets you will find in the enthusiast community in various special colour schemes are in Cherry's profile actually, but often with options for mounting on either Cherry-brand or "normal" keyboards.
Personally I prefer the slightly shorter keys.

In the image posted above you can see Tai Hao - which is practically OEM profile. Note especially the keys on the Function key row that are extra high - keys in that size do have some noticeable wobble which you may not like.

[CapnHighlands]

I would say that it's personal preference, like most things in this hobby. Signature Plastics has a new profile called G20 that is basically completely flat. They're actually pretty nice to type on but I think they're a bit ugly. For what it's worth, I prefer DSA profile which is pretty short.

The only thing I would say tall caps can contribute to is definitely some wobble, especially on Gaterons. I have a non-sculptured set of SA keycaps on Gateron Blacks and there is some decently noticeable wobble.