What is the difference between :

konuşuyorum and konuşurum?

The difference simply is:

konuşuyorum - I am speaking 

konuşurum - I speak

we generally use simple continous tense instead of simple present tense.

so if i want to say that for example as a hobby i read the newspaper is it better to use gazeteyi okurum or gazeteyi okuyorum?

I read newspaper every day.

Ben her gün gazete okurum.

Ben her gün gazete okuyorum.

Both has same meaning.

But if you don´t use "every day". It is better to use present tense.

I read newspaper.

Ben gazete okurum.

But I must say without "her gün" meaning is obscure.