What is the difference between the two formulas V=E-Ir and V=IR. According to the former V is inversely proportional to I (current) while according to the other one V is directly proportional to I.

Both refer to terminal potential difference. the first formula V=E-Ir refers to the potential across the ENDS of the battery when a current I is flowing through the battery. the internal resistance r of the cell uses up a part therefore terminal potential difference V decreases as I increases because the factor Ir increases.

. V=IR is the potential difference across an external resistance R. The greater the current, more is the IR factor hence greater is V across the resistance.

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