What happens if we improve the time hierarchy theorems? In a nutshell, the time hierarchy theorems say that a Turing machine can solve more problems if it has more time for computation. In detail for deterministic TM and time-constructable functions $f,g$ with $f(n) \log f(n) = o(g(n))$ it is
$$ DTIME(f(n)) \subsetneq DTIME(g(n))$$
and for nondeterministic TM and time-constructable functions $f,g$ with $f(n+1)=o(g(n))$ it is
$$ NTIME(f(n)) \subsetneq NTIME(g(n)).$$
There are a lot of (old and current) results which use the time hierarchy theorems to prove lower bounds. Here are my questions:


What happens if we can prove a better
result for the deterministic or nondeterministic case?
If we can prove that there is a gap
between the deterministic time
hierarchy and the nondeterministic
time hierarchy, does this imply $P \neq
   NP$?
About your second question. No, that would not imply $P \neq NP$. Hierarchy theorems are mostly useful to determine the amount of a single resource needed by a TM so that additional problems can be solved.

For example, we know that $DTIME(n) \neq NTIME(n)$. Let $f(n) = n$, $g(n)$, $h(n)$ such that $f(n+1) = o(g(n))$ and $f(n)log(f(n)) = o ( h(n) )$. 

From the hierarchy theorems it follows that $ DTIME( f(n) ) \subsetneq DTIME( g(n) )$ and $NTIME ( f(n) ) \subsetneq NTIME( h(n) )$.
 Under those assumptions, $ NTIME( g(n) ) \subseteq DTIME( h(n) )$ is possible.

The hierarchy theorems can be used to determine relationships between resources, given an equality between them. For example , assume that
$ NTIME(2^{n}) = SPACE( n )$. We know that $NTIME( g(n) )$, for $g(n)$ such that $2^{n+1} = o(g(n))$, cannot be equal to $SPACE(n)$ , due to the NTIME hierarchy theorem.