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I am confused whether to use Lagrange's Mean value theorem (LMVT) or normal Mean value for integrals to find the mean value.

For example :- $f(x) = x^2 + 3x + 2$ in the interval $[1,4].$

I get the value of the mean as 2.5 using LMVT and value of mean or average value of function as 2.593 using mean value for integrals method.

Why am I getting different values ?

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You say you want the "mean or average value". That is \begin{align} & \frac 1 {4-1} \int_1^4 (x^2+3x+2)\, dx = \frac 1 3 \left[ \frac {x^3} 3 + \frac {3x^2} 2 + 2x \right]_1^4 \\[10pt] = {} & \frac 1 3 \left( \frac {64} 3 + 24 + 8 \right) - \frac 1 3 \left( \frac 1 3 + \frac 3 2 + 2 \right). \end{align}

Neither of the mean value theorems you cite gives a way of finding the mean value. Lagrange's mean value theorem is about the mean value of the derivative. That mean value is expressed as a difference quotient of the function that gets differentiated.

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