Hicks: He will be a dogmatist but not a mere sceptic;

Yonge: he will pronounce dogmas, and will express no doubts;

Mensch: He will assert his opinions and will not remain in doubt about them.

I prefer Mensch's here.

The original text is simply two verbs: δογματιεῖν τε καὶ οὐκ ἀπορήσειν. The contrast is a choice between being a "dogmatist" or a "skeptic." 

• δογματιεῖν (dogmatiein) is, of course, related to English "dogmatic, dogma, etc." but δογματίζω  the verb itself, per LSJ, is just "to lay down as an opinion." The Ancient Greek dogma itself means "that which seems to one, opinion or belief."

• ἀπορήσειν (aporesein), per LSJ, is "to be at a loss, be in doubt, be puzzled." This is the same word as used in talking about when the sage will make money and is translated in some texts as "to be in poverty/need" so each instance could be translated as "to be in want."

The words are also used in other places. In the letter to Herodotus:

[37] In the first place, Herodotus, you must understand what it is that words denote, in order that by reference to this we may be in a position to test opinions, inquiries, or problems (ἀπορούμενα), so that our proofs may not run on untested ad infinitum, nor the terms we use be empty of meaning. [38] For the primary signification of every term employed must be clearly seen, and ought to need no proving; this being necessary, if we are to have something to which the point at issue or the problem (ἀπορούμενον) or the opinion before us can be referred. 

Diogenes Laertius calls Epicurus (10.11) the man who laid down (δογματίζων) that pleasure was the end of life. He also uses the word δόγματα (dogmata) which is often translated "doctrines," but I would suggest something colloquial like "what Epicurus laid down."

With those two options available, being a dogmatist or being a skeptic, it seems to me that the significance is that one path leads to declaring that knowledge can be known, that it is possible to "take a stand" on what can be known about reality. The other path leaves one "puzzled," "in want of knowledge.," or simply letting problems remain without resolving them or at least proposing solutions. The second path implies that we can't really know anything. Epicurus was opposed to this idea wholeheartedly.