CONFIG_HZ_1000 vs. CONFIG_HZ_250

You have increased the chances of losing timer ticks to the clock,
messing up your system time.

Yes, you increase the frequency at which the scheduler kicks in.
That in turn means that if there is more than one runnable process
more context switches and more TLB and cache flushes.  I wouldn't
imagine that it would amount to a particuarly significant increase
in overhead but it is an increase in overhead never the less.

Having said that I'm reading this in c.a.e. and am not really a
Linux expert.  I'm more of a BSD man myself so I can't really steer
you down a better avenue.

If all software is done according to the most recent standards, normal
software should not notice the difference. There even is a HZ setting
for Kernel space and a different one for user programs, so even user
programs that rely on the standard setting, instead of using the HZ
constant for timing issues, should work as expected. The faster
interrupt rate of course might reduce the overall system performance,
while increasing responsiveness for soft realtime applications.

-Michael