Jonathan Marks explains the difference between mood and modality.
So what about modality, then?
A basic distinction in our use of language is that we sometimes state facts, like It's raining, It hasn't rained for weeks, and we sometimes express personal perceptions, opinions and attitudes, such as I think it's going to rain, We could do with a bit of rain, It must have been raining, Let it rain – I don't mind getting wet. The term 'modality' includes various types of such personal perceptions and attitudes. A broad categorisation of the main types of modality would be:
1. Possibility, including ability and permission
2. Necessity, including obligation
Another important division, which cuts across these, is that between extrinsic and intrinsic modality. Extrinsic modality refers to our assessment of the possibility, likelihood or necessity of situations and events:
It can be chilly here at night.
You must be Dr. Livingstone.
You might have left your coat at the pub.
Intrinsic modality refers to people's control over, and evaluation of, situations and events - whether these are permitted, desirable, approved of, etc.:
You can't park here.
You must be careful.
This door shouldn't be left open.
One major way in which we express these meanings is by using modal verbs. These are familiar items which tend to be treated quite thoroughly in courses and grammar reference books with good reason, because they're certainly important, and there's quite a lot to learn about their forms and meanings, and their functional uses in advising, suggesting, directing, asking for and giving permission, etc. The main modal verbs are may, might, can, could, must, will, would, shall and should, and most writers also recognise 'semi-modals' or fringe members of the group, such as have (got) to, ought to, need, and dare, which in terms of form behave in slightly different ways, but express similar types of meaning. Apart from modal verbs, though, there are plenty of other ways of expressing modality:
Maybe you left your coat at the pub
I'm quite capable of looking after myself.
We weren't allowed to leave our luggage at the hotel.
You'd better get a move on.
What do you think they're going to do?
How am I supposed to open this?
We're likely to need some more time.
Would you be willing to help us a bit?
Dr. Livingstone, I presume?
It's essential to give yourself plenty of time.
Looking at mood and modality together
You could say, for example, that:
This is the oldest building in the town demonstrates declarative mood and no modality.
I mustn't forget my keys demonstrates declarative mood and intrinsic modality.
Is it possible to enrol for the next semester yet? demonstrates interrogative mood and extrinsic modality
But how useful are these terms?
You say that you need to explain the difference between 'mood' and 'modality', but you don't say who you need to explain them to. I don't think 'mood' is a very useful term for learners of English. For one thing, it's confusing: it's got nothing to do with the more familiar use of the word 'mood'. For another thing, it's very abstract and probably doesn't help learning. It's probably more effective to stick to more familiar terms like 'statement', 'question', and 'imperative'. Personally, I wouldn't use the term 'subjunctive' with reference to English; I'd only use it in teaching learners whose own language has specific subjunctive verb forms, to point out that English doesn't have such forms, but sometimes uses unexpected verb forms to do the same job.
'Modality' is also pretty abstract, but maybe helpful as a reminder that modal meanings are carried not just by modal verbs, but in other ways as well. For learners, it's probably most useful to learn the various meanings and functional uses of modal verbs and modal lexical expressions: making logical deductions, asking for and giving/refusing permission, making requests, talking about ability, offering to do things, making predictions, and so on. And this is the approach usually taken in coursebooks. At a more advanced stage, it might be useful to do some review and extension work on ways of expressing possibility, necessity, volition, etc. in English, possibly drawing comparisons with the learners' L1 if this seems likely to be fruitful.
As always, the priority should be learning to understand and use the language itself. If terminological labels help, so much the better. But there's a lot of unhelpful terminology inherited from grammars of classical Greek and Latin which we could really do without; I think 'mood' is a clear example of this, while 'modality' is more of a borderline case.