If they are the same syntactically, why is it right to say 食べている but not 食べいる? Why is it right to say 食べ続ける, but not 食べて続ける? Forgive my ignorance on this topic.
The te-form and the masu-stem (aka i-form) are not the same syntactically. There are cases where they seem interchangeable, but in most cases, they are simply different. For example, only the masu-stem can work as a noun. You must use the te-form before a subsidiary verb (eg 飛んでくる, 歩いてみる), and you must use the masu-stem in a compound verb (eg 飛び出す, 歩き始める).
In English, gerunds and infinitives sometimes seem interchangeable. "I like to play tennis" and "I like playing tennis" have the same meaning. But you can say "I want to play tennis" but not "I want playing tennis". It's simply because gerunds and infinitives are syntactically different.
When joining two clauses, 連用形 sounds more formal and literary. For other uses of 連用形, please see: Do I have a good grasp on the basics of what the continuative form is?