One important difference is that 겠 indicates more reflexive and spontaneous thought or feelings, like "it might/would ...", whereas 것 같다 involves more thinking and thus sounds more indirect, like "it seems" or "it appears".
For example, if a baby is about to fall from a place, you'd say 저 애(가) 저기서 떨어지겠다 rather than 떨어질 것 같다. Both are correct, but 겠 sounds more urgent and visceral while 것 같다 sounds like a leisurely judgement.
Another example would the colloquial "be dying (from pain)", which translates to (아파) 죽겠어 or (아파) 죽을 것 같아. Again, both can be said, but 겠어 is more common.
As for the tenses, you're right that 것 같다 can have wider applications.
Here are the main possible forms.
- 갔을 것이다 (would have gone) / 간 것이다 (went) / 가는 것이다 (goes; is going) / 갈 것이다 (will be going).
- 갔겠다 (might have gone; perhaps (they) went) / 가겠다 (will/would/might go).