Suckling is only used with a teat - it means milk feeding from a mother. Usually you don't mention what it's 'on', as that is obvious. Sucking can be on anything. So, with in the specific case of a teat, they are nearly synonymous. Obviously - in your phrase, there are some unsaid implications! The phrase 'suckling pig' means a piglet that has not yet been weaned.

The explanation of the metaphor here is that the speaker is dependent on TV from the established 6 stations. If he gets a satellite dish, he won't be dependent any more.