I am going to assume that you connected to the Sync service using your Firefox Account, which is used for a number of Mozilla / Firefox services. And your question is about Sync not working or completely working for data transfer to the new computer.

Did any type of Firefox data 'land' on that new computer when you connected it to Sync with your Firefox Account?

Or is everything else there and just browsing history and save logins are gone?

And as far as saved logins, were you using the Master Password feature on the Linux system? If so, the Master Password will be needed on the new computer.

No anything is not synced. And in my linux system I was not asked to enter Master Password....My life has become tough after this...I do not have any idea how to recover this... Help me.

No, you're not going to be asked for the Master Password on a device that where a Master Password wasn't enabled.

If nothing was synchronized, I have to ask if you reset your Sync password when attempting to log into Sync on the Windows computer?

One more thing; what version of Firefox were you last using on that Linux device? Could it have been an old version, like Firefox 28 or earler? Or how long ago? Could it have been a couple of years ago?

It was synchronized on my previous computer. Yes I changed password as I tried 2-3 times and I got wrong. But that has nothing to do with my data. If I have my account and I can reset password as well why can I get my data back?

Firefox on last machine was the latest one. And in this machine too.

Keep in mind that I am just another Firefox user. I don't work for Mozilla nor I am a software developer of any type - just a user who was enthralled with Sync when it was first announced back in Dec 2007 and have been "playing with it" since then.

Sync uses dual encryption (kA & kB) "keys" which are generated from the account Password that is in use at the time data is sent to the Sync server. Sync data is encrypted as it leaves each device; remains encrypted while traveling to and from the Sync server and while sitting on the Sync server; and is decrypted when it arrives back on a device that has Sync setup using the original Password and the same dual kA & kB encryption "keys". Mozilla never sees the account Password, it handles the dual "keys". No backup of the old data is saved; there are no backups at all.

When the Sync Password is changed, the account data is cleared from the Sync server. please see the Reset warning screenshot that I posted A different Password will create different kA & kB "keys" which won't decrypt the previously sync'd data, thus rendering that data useless.

That "clearing" action also performs a maintenance function as of the Firefox 29 / Sync 1.5 version; allowing the user to clear corrupt data that is "stuck" in Sync. With the earlier Sync 1.1 version if Sync data was corrupted the user had to create a new Sync account using a different email address. Thru the "run" of Sync 1.1 in Firefox 4.0 thru 28.0 I ended creating 7 different Sync accounts as I was testing Sync to see what would break it. I actually had to create a few new email accounts just to keep testing Sync.

tl;dr version: Sorry about your loss of data.

Changing the account password was a fatal mistake for your data; it was gone with the first password reset.

Improper usage of Sync which isn't intended as a backup or storage medium. Sync was created to synchronize Firefox data across multiple devices as each device is being actively used.

So what can I do now? Is there any alternative?

There's nothing you can do if you changed your Sync password, your data is gone with no way to recover that data.

If I do login from the same system once again than do I have any chance to get the data back?