I like the ability to show trailing whitespaces, so I have this setting.
set list listchars=tab:\ \ ,trail:·
I never use tabs (although I don't think tabs are bad, it's just a convention), but sometimes I'm reading someone else code and they use tabs. I don't want to see tabs as some special characters in that cases.
But with the settings above vim would show tabs with different background colour. I don't want this. Can it just show trailing whitespaces and leave everything else alone?
You can omit tab from listchars, but that will cause vim to fall back on the default (showing tabs as ^I). So using tab:\ \ as you have is a reasonable way to keep them "invisible" when list is set.
The characters shown with list are highlighted with the SpecialKey or NonText groups, depending. Tabs, in particular, use SpecialKey, so it sounds like your color scheme has a background color for that particular group (see: :help listchars for more).
Unfortunately SpecialKey is used for trail as well, so whatever changes you make there for tab will apply to trail as well. If you go this route, you will only be able to change the foreground color of SpecialKey, really, without visibly impacting the tabs.
One alternative might be to avoid using list at all; in your .vimrc:
syntax match TrailingSpace /\s\+$/
hi TrailingSpace guibg=red
for example, will highlight trailing whitespace with a red background independent of whatever you have going on with list.
'listchars'isn't supposed to change background. You can for sure try to droptab:\ \part, but I wouldn't expect it to change anything (that's the default). Highlighting of tabulation might come from a different source. You can run:for id in synstack(line('.'), col('.')) | echo synIDattr(id, 'name') | endforwith cursor on a tab to see if there is any extra highlight group defined.