I am a native speaker of Japanese but not of English. So my English may not be natural but I hope to be helpful.
たぶん and でしょう have similar meanings but are different parts of speech.
たぶん（=おそらく）'s definition is:
1 〔おそらく〕probably; perhaps; possibly; maybe
So たぶん is an adverb like probably, perhaps, or possibly. And たぶん’s meaning varies from probably to possibly according to contexts.
でしょう（=だろう）’s definition is:
(1)〔話者を主語にして〕I think [suppose/guess] (that)....
推量の助動詞 means can be, may be, could be, will be, etc.
So でしょう is an (auxiliary) verb like think, suppose, guess, will be, or may be.
Therefore you can say:
明日はたぶん雨だ。 (It probably rains tomorrow.)
明日は雨でしょう。 (It will rain tomorrow.)
明日はたぶん雨でしょう。 (It will probably rain tomorrow. Or I think it probably rains tomorrow.)
(I quoted the definitions from プログレッシブ和英中辞典)