If you let someone down, you disappoint them in some way. You did not do something which you had promised to do, or you did not behave as you were expected to do, and the other person feels disappointed or annoyed. 'Let down' can be used in many different situations.
By contrast, 'stand up' refers to only one specific situation. If a man or woman has arranged to go out on a date with you, and they don't turn up, we say that they stood you up. The person who waits has 'been stood up' by the person who fails to arrive.
As you can see, the two iexpressions are related, but they aren't used in the same situations.