What is the difference when a jutsu is performed by weaving only some of it's seals and when performed by weaving all of them? Well, I noticed that plenty of times ,for instance Itachi performed the great fireball jutsu sometimes with one hand seal and some other times with all the ones necessary
Weaving signs is a chakra manipulation technique that enables you to gather it in specific ways to execute specific techniques based off the chakra nature of the individual. 

These signs are defined as seals and can be read about in the seals wiki in narutopedia. 

As quoted in the wiki :


  While techniques may require a number of hand seals to work, a skilled ninja can use less or even one to perform the same technique. 


It is also noted that some shinobi can master a technique so well that they can perform instantaneously without weaving signs but this takes years of training or an abnormally strong level of chakra control. 

For instance Sasuke and chidori in the shippuden series is a common example of mastery as he learned to perform the technique seamlessly without needing time to weave. 

Minato was also known for this with the flying raijin and rasengan. 

As a shinobi masters a technique they are able to use less seals, one hand, and even no seals.