Let \(K\) be a global field and denote by \({\mathcal M}_K\) the set of pairwise inequivalent absolute values \(v\) of \(K\) satisfying the sum formula \(\sum_{v \in {\mathcal M}_K} \lambda_v v(z) = 0\) for all \(z \in K^*\) with multiplicities \(\lambda_v\). Let \(E: Y^2 = X^3 + aX + b\) be an elliptic curve over \(K\). For \(v \in {\mathcal M}_K\) and \(z \in K^*\) define \(v(z)= -\log |z|_v\). Set \(\mu_v = \min \{ v(a)/2, v(b)/3 \}, \mu = - \sum_{v \in {\mathcal M}_K} \lambda_v \mu_v\) and \(\alpha = - \sum_{v \in {\mathcal M}_K} \lambda_v \alpha_v\), where \(\alpha_v = v(z)\) if \(v\) is archimedean and \(0\) otherwise. For \(P= (x,y) \in E(K)\) the modified Weil height is defined locally by \(d_v(P) = - \min \{ \mu_v, v(x)\}/2\) and globally by \(d(P) = \sum_{v \in {\mathcal M}_K} \lambda_v d_v(P)\). Finally, let \(\widehat{h}\) denote the global Néron-Tate height on the group \(E(K)\). The purpose of the present paper is to show that by applying simple formulae on \(E(K)\), one can obtain an already usable estimate for the difference \(\widehat{h} - d\). Moreover, if \(K\) is an algebraic number field the function \(d\) may be replaced by another modified Weil height \(d_{\infty}\) arising from \(d\) by just taking the archimedean absolute values of \(K\). More precisely it is proved that if \(P\in E(K)\) then we have \(-\mu - 5\alpha /3 \leq \widehat{P} - d(P) \leq 2\alpha /3\). If \(a=0\) and \(K=\mathbb{Q}\) then a comparison with earlier results of \textit{J. H. Silverman} [Math. Comput. 55, 723-743 (1990; Zbl 0729.14026)] and \textit{S. Siksek} [Rocky Mt. J. Math. 25, 1501-1538 (1995; Zbl 0852.11028)] is given.