Generally, apps need to be actively installed on a different phone, although the phone may be able to sort out some of them for itself if they are on the SD card. Also, presumably some of the apps might require a different version for the different android version.
If you go your Google account, then select the Dashboard, you'll see your Devices; if you haven't already done so, you will need to add your G510 here. I've only got the one device, so I'm not 100% sure what happens next, but I think it should then be possible from within Google Play: My Apps to install the missing apps on your new phone.
If you have any apps which can only be installed on one phone, I guess you may have to uninstall them from the G300 first, but perhaps someone else can advise on this?