For example if I play staccato there is a silence between each note.
What is the technical term for that silent part or silence duration?
Note gap?
There is no standard technical term. In addition to the words mentioned in the other answers, I frequently hear metaphorical terms such as "space" or even "air." Depending on the context, I think I would most likely use "space" or "gap." If discussing the concept in detail, I might use a phrase such as "time between the notes."
With most instruments, and where they're played, there's actually some reverberation sound in between short notes. Not all strings are damped on pianos, the body of guitars continue to vibrate in sympathy, the room will have some echo effect after a short note is stopped, when just about any instrument is played staccato. So, rarely is there genuine silence. Maybe it would sound odd, certainly different, if there was.
Music is made up using sound and 'silences' - aka rests, and rests will be the term closest. Although when staccato is written, the rests are implicit - not noted, so to speak.
Pause isn't appropriate, as that invokes a change in tempo, which rests certainly do not. Gap, or space would do, and we talk about note separation, but nothing more specific comes to mind.
I guess, there is just the process producing that "gaps", which has a name: articulation, the most frequent ones are:
Update: some kinds of articulation address other properties as well, e. g. marcato will also influence the volume or emphasis.
If you really have to talk about the gaps, articulation would not be the appropriate means to notate it (since there is a big range of impact by the interpretation of the artist), you would then explicitly shorten the note duration and insert rests, see this question: Notation - when to use staccato vs rests.
When playing staccato, or non-legato the focus is on the notes being separated rather than on the gaps that separate them. There is no special name for them.
If we however request the silences to have some very specific duration, then perhaps rests should be used instead of staccato or non-legato notation.
Depending on the context you can use various names. Gap, silence... One can also speak of notes being separated more or less.