What problems did Arabs and Israelis face with the independence of Palestine?

This question is a little hard to understand.  You say
"did" but Palestine is not independent (if you're talking about the Palestinian state
that is supposed to get made out of the West Bank and Gaza).  Israel did become
independent, but it was not called Palestine after
independence...

I'm going to assume that you're talking
about the time when Israel became a country in 1948.  At that point, the Israelis faced
the problem of being ringed with countries that hated them.  Various Arab countries
attacked Israel as soon as it became independent.  Because it was ringed by enemies,
Israel was ina precarious situation.

The Palestinians faced
the problem of having been driven off their land.  Most of them had chosen not to become
Israelis.  They had to figure out how and where they would live now that their land had
become part of Israel.

Since then, of course, the two
groups of people have continued to face these problems which still exist to this
day.