GOG Galaxy or off-line mode?

Tell me which is better, if you want to tell me what is the difference

Well, if you download the offline installers, you can store them everywhere you want and can install the games without internet connection.
Galaxy needs an internet connection and tracks your gaming time and makes it visible for others. It`s also needed if you want to use archievements, which some games have.

Conan, what is best in life?

Galaxy is better for giving you Cloud saves, Achievements, telling you how long you've played a game, and providing a one-stop place to quickly & easily launch any of your GOG games.

Offline installers are better for archiving your games as backup copies in case GOG ever goes out of business or if you lose access to your account at some point in the future.

I could tell you... but then I'd have to kill you!

Well, technically I wouldn't HAVE to, but I think I would anyway, for shits and giggles.

How is babby formed?

You can use both at the same time.

If you have enough HDD space and a less-than-stellar internet connection the quickest way to install a game is to use the offline installer (which you hopefully have downloaded some time in the past) and then you can still use Galaxy to launch the game for time tracking, achievements and cloud saves, or even MP.
Best of both worlds.

"Hot water, good dentishtry and shoft lavatory paper."

What's up with this guy?

I'm going to say what has already been said but if you have difficulty deciding between the two then download the offline installers of your games and use Galaxy to play your games for convenience.

Oranges are better than apples.
Every other choice is just subjective.

In the end, it doesn't even matter

There's many more "help my galaxy game doesn't work" than "help my offline game doesn't work" throughout this forum.

I've put my trust, in you, pushed as far as I can go

People against DRM, use:

GOG Galaxy:

For downloading game files/extras (the offline installers).
For manually (better manually than automatic) updating their games.
In game time? ONLY for multiplayer functionality.

OFFLINE INSTALLERS:

For installing your game.
For playing your game.
For burning them into storage media.

People acquainted with Steam, clients, etc:

Use GOG Galaxy, for all those shiny perks and useless features (game profile, time played, achievements, cloud save, etc)