As indicated by narx's answer to a similar question on ELU (English Language & Usage, ELL's "sister site" aimed primarily at linguists, etymologists, and serious English language enthusiasts), only little can be used as an unqualified noun to mean a small amount (where adjectival small needs an explicit noun, such as amount or quantity).
The other interesting point made about little there is that in...
1: I have a little experience teaching English.
2: I have little experience teaching English.
...the two sentences have almost opposite meanings - #1 implies you have at least some experience, whereas #2 implies very little (perhaps almost none). In fact, #1 is often used somewhat facetiously to imply you actually have a lot of whatever is being spoken of.
Apart from the above distinctions, there's also (as kiamlaluno points out) the fact that little often carries more "affectionate" connotations than small. And from BBC Learning English, little is more common in the sense not much when coupled with abstract nouns such as hope, chance, change, effect, use and point.
In most other contexts, there's little to choose between the two words. Use whichever you like.
I have no room for other things in my mind.
I have no space for other things.
Both of these are figurative speech, and they are acceptable as is. One might add the word extra or spare as well:
There's no spare room for anything else in my mind.
I have no extra space for new things in my brain right now.
(I decide to change the way you said other things, because it's hard to figure out what "things" are in your mind already, and what other things you don't have any room for.)
But, yes, you can use words like space when talking about the brain for the mind. In fact, the word attic is sometimes used to refer to the mind metaphorically, because it's in the upper part of our body, and it's used to store things. Doyle used this analogy in 1887, in a Sherlock Holmes story:
“I consider that a man's brain originally is like a little empty attic, and you have to stock it with such furniture as you choose.” (Sherlock Holmes, A Study in Scarlet)
As you guessed, words like space and room can also be used for a schedule, as well as a brain:
I'm sorry, I'm too busy; I don't have room to meet this week. Can we try for next week? I have more free space on my calendar then.
In addition to brains and calendars, you could also use such words to describe, say, a person's frustration level. For example:
My car won't start! Dammit! I don't have room for this right now.
might mean that the speaker is under a lot of stress, and this isn't a good time for his car to break down. Here's another example from a book:
As much as I don't have room for this in my life right now, I can't help but be intrigued. “What kind of plan?” I ask. (N.L. Shepherd, Stealing Bases)
Best Answer
To me, room means a space with specific dimensions. As Damkerng T. noted, it implies that the space is sufficient for a certain purpose.
Means: Clear a space of sufficient dimensions for me to sit on.
In this example space is being used to refer to a space of geometrical dimensions. You could not use the word room here, which is defined as an extent of space occupied by or sufficient or available for something.
To further clarify, take a look at the following example including both the terms space and room:
^ no dimensions specified, so far it could mean a space of any size
^^ room for our car clarifies that the the space must be big enough to fit an object of size X.
Also, you can use the word "spot" (a particular place, area, or part) to refer to a space that's of sufficient dimensions for something.
Spot implies that the space needs to be big enough to fit a certain object as well.