Ok. I eventually got my SMS back. But if someone is searching for an answer, here is my solution that, I hope, will help.
I guessed the problem came from the android version. So, basically, I did the following:
1. Grap your old phone under Android 4.0.4 (you can use someone else's phone, just make sure to backup their messages before you do anything. To be (almost) sure THIS backup will work, I recommand "sms backup +" or "sms backup and restore")
2. Now, install Helium and backup their SMS with it. It will create a backup file. (it should be inside a folder called "carbon" and sould be named "com.android.providers.telephony")
3. Then, take your old backup (same name under same folder in your new phone), and replace the one you just created at step 2 with it.
4. Click restore, and all your sms should be back (if you've taken someone else's phone, you should delete all their sms, otherwise yours and theirs will be mixed. Again, make sure to backup everything before)
5. Then, backup these sms freshly restored with the app downloaded at step 1. I recommand to use the Gmail backup, so you don't have to travel your backup file from one phone to the other with your PC, etc... but that's your choice.
6. Finally, install the same app as step 1 on your new device, restore your sms with the backup file you just created, and that's it!
I hope this little explanation is clear enough, it is waaay simpler than it seems (because I tried to cover most cases). If you encounter problems, leave a comment below, and I'll see if I'm able to answer (I don't promise anything okay? :p).
PS:Again, I hope my English isn't too bad, but I tried my best, if you see mistakes, just tell them me in the comments section :)