King Reza: Another shot at it from a non-native:

1. Probably it'll rain and It's likely to rain. are similar, I believe, and indicate a very strong possibility that it will rain. The sky is probably very cloudy as you say that.
2. It may rain., Perhaps it'll rain. and Maybe it'll rain. indicate that you believe that is some possibility for rain. Maybe in a decreased degree of confidence in that probability, though I'd say they may be used interchangeably.
3. It might rain. a bit like the previous one. But may also indicate a wish, like "I wish it would rain, the plants are thirsty and rain would help".
4. It'll possibly rain. doesn't sound right to me, especially with that contraction. But I'd say it accounts for a strong possibility too...