Learn English – Should I use “figure” or “figure out” in the following sentence

The meanings are very similar. The only difference, to my mind, involves the effort involved.

You can think of something out of the clear blue sky, without even trying. If you figure something out, you have set yourself a problem, or been set a problem, and solved it; this implies more effort. To work something out also implies an effort, and probably a more prolonged one; you can figure something out quickly, but it usually takes time to work something out.

For most practical purposes, though, they can be used interchangeably.

You're right that definition-wise the phases do mean the same thing, however they convey different subtleties. Adding "out" after the verbs you mentioned has a tendency to sound all-encompassing, a complete action, and sometimes pompous (in the case of "spell out"). Look at the following sentences:

Jerry planned the trip.

Jerry planned out the trip.

In the first sentence, the idea conveyed is that the trip is planned, Jerry knows where he's going on his trip and probably what hotel he will stay in. The emphasis is on the planning process being completed.

The second sentence gives the idea that Jerry planned out every last detail of the trip down to the minute and not only does he know where he's going, he's know the exact hotel he'll stay at, the exact museum he's visiting, and where he will eat his lunch. The emphasis in on the extent of planning that Jerry has put into his trip.