Note: This answer is anecdotal, as it is based purely on my personal experience. Feel free to edit if you have studies or well known principles to support our undermine my point.
TL,DR; Do not mix warm and cold greys, of you do not want a "muddy" look.
What I've noticed during my work as an illustrator is that colour choice is often more about relative contrasts than about the absolute colour on its own, and this is the case for aspects like size, saturation, value, composition etc. too.
So let's look at the context:
- If your work only contains desaturated and middle-to-low value colours, you can work with each of the greys as you would with the saturated, lighter version of those colours (all scaled up linearly). It'll probably be easier to work with bright colours, see what fits, and then dull it down. This will have other problems of low saturation image, but that's not in the scope of this question I believe.
- What's more likely is that your work will contain at least one stronger colour*, as you'll want to have legible text and maybe some accent colours too. This will introduce the strong contrast commonly needed for accessibility and vivid experiences. It will also put your background in context with the text and accent colours. There's something like an uncanny valley when it comes to contrasting colours: choose very similar ones, and they seem like a slight variation of the same colour; with huge differences you have a clear cut contrast. Choose a colour relation in-between, and you'll likely get something that's neither a smooth variation nor a clean contrast, but a composition that feels undecided and muddy.
Now this might be something you're going for, if you aim for such dissonances as a stylistic tool to draw attention to your work. Since you asked this question on UX SE though, I believe you are interested in a "nice" user experience, without artistic dissonances. In that case I'd strongly recommend against using two very similar but slightly different hues of grey.
*I'll count Black and White as a colour here for the purposes of this question