I would be tempted to say that “au plus” or simply “plus” seems more correct to say “the more”, as in “au plus je grandit”/”plus je grandis” to say “the more I grow”.

In case you do stumble upon one of the examples you describe, an easy way to differentiate them without the full context is to change them to the plural form.

“The most” will still make sense in the plural:
Le plus grand -> Les plus grands (correct)

While “The more” will not:
Le plus je grandis -> Les plus je grandis (incorrect)

“The most” is a superlative, in French, un superlatif. “Il est le plus grand d’entre nous” = he is the tallest of us.

The more is a comparative, in French, un comparatif. “Il est plus grand que moi” = he is taller than I.

The use of “le/la”, as Nico Mezeret said, is necessary linked to a superlative. See : http://la-conjugaison.nouvelobs.com/regles/grammaire/le-superlatif-141.php.

So, there is no potential mistake: “le plus grand” is the tallest!