Is there some way I can get vim's colorscheme rendering more like gvim's?

Yes! As long as your terminal supports "true color," you can ask Vim to use the gvim color settings of the colorscheme by enabling the 'termguicolors' setting:

When on, uses highlight-guifg and highlight-guibg attributes in the terminal (thus using 24-bit color).

Requires a ISO-8613-3 compatible terminal. If setting this option does not work (produces a colorless UI) reading xterm-true-color might help.

Simply add this line to your vimrc:

set termguicolors

If that doesn't work, take a further look at the documentation to ensure your terminal supports this feature and is properly configured to use it.

Is there some way I can get vim's colorscheme rendering more like gvim's?

Yes! As long as your terminal supports "true color," you can ask Vim to use the gvim color settings of the colorscheme by enabling the 'termguicolors' setting:

When on, uses highlight-guifg and highlight-guibg attributes in the terminal (thus using 24-bit color).

Requires a ISO-8613-3 compatible terminal. If setting this option does not work (produces a colorless UI) reading xterm-true-color might help.

Simply add this line to your vimrc:

set termguicolors

If that doesn't work, take a further look at the documentation to ensure your terminal supports this feature and is properly configured to use it.

Your terminal emulator, MobaXterm, is supposed to support "true color", so it should be possible to make it work.

If setting TERM=xterm-true-color is giving you trouble, then try to keep TERM=xterm-256color and set vim options t_8f and t_8b explicitly, as recommended in :help xterm-true-color:

let &t_8f = "\<Esc>[38:2:%lu:%lu:%lum"
let &t_8b = "\<Esc>[48:2:%lu:%lu:%lum"

(And, if that doesn't work, try using semicolons instead of colons in those expressions.)