I'm in India right now and I'm planning to go to the Himalaya. I could
either go to Leh, Ladakh, India or Kathmandu, Nepal.

I just have 4 days to spend and therefore would like to get the maximum out of it.

As of now I am more interested in Kathmandu (mainly cultural reasons)
but I fear that the monsoon would hinder my travel or sightseeing
experience there so I'm actually thinking of going to Ladakh/Leh to
have a sure bet on the weather.

Can someone tell me to what extent sightseeing/trecking will be
limited right now in Nepal due to the weather? (How much rain is
there? When does it rain (at night/during day) Is it all foggy/cloudy?
etc.)

Issues such as price of flights/duration of travel to get there don't matter.

I'm no expert, but Kathmandu itself is only in the foothills, and here
is the weather forecast, about what I expected:
http://www.wunderground.com/global/stations/44454.html
Whoops, that was a German site, but here to compare:
http://www.qwikcast.com/cgi-bin/forecast.cgi?zip=Kathmandu,NP
Note the rather moderate lowest temperatures, Kathmandu is not in the mountains.
If there is a 100% chance of rain, it will be monsoon overcast.  If it
is anything like what I remember:  soggy, 100% humidity, even if the
temperature is refreshing after you have come up from India.

The weather in Leh is much better (I think):
http://www.qwikcast.com/cgi-bin/forecast.cgi?zip=Leh,IN
http://weather.cnn.com/weather/forecast.jsp?locCode=IN072

Ladakh also has the advantage that you don't enter another country.  I
don't know how problematic that is for a short visit from India to
Nepal for you (US?), but Nepal is sticky about maintaining its
distance from India.

Leh is definitely the smaller place, which I would find more
interesting.  But I have been to Kathmandu, which is also interesting,
but quite a city now with many tourists.

I am sure you can find all the tourist/trekking information you need
for either place.

If you have to depend on my suggestions, I sincerely hope that you
won't be disappointed and wish you a excursion.