The simple answer is that they don't.
Here ut and ne introduce something akin to a purpose clause:
timeo ut veniat!
Literal: 'I fear so that he may come!'
Idiomatic: 'I fear that he may not come!'
timeo ne veniat!
Literal: 'I fear so that he may not come!'
Idiomatic: 'I fear that he may come!'
Both the Latin and the literal English contain a purpose clause, introduced variously by ut, ne and so that. By contrast, the idiomatic English contains indirect speech, introduced by that (mark the absence of so).
So really, it's a different construction altogether. It also exists in English - we just don't happen to use it after verbs of fearing.
