There is a difference in the behaviours of two geometric evolution equations that otherwise show a lot of similarities: the harmonic map heat flow and the Yang-Mills heat flow. Equivariant solutions in the critical dimension can blow up for the former flow [\textit{K.-C. Chang, W.-Y. Ding} and \textit{R. Ye}, J. Differ.\ Geom.\ 36, 507--515 (1992; Zbl 0765.53026)], but they do not for the latter [\textit{A. E. Schlatter, M. Struwe} and \textit{A. S. Tahvildar-Zadeh}, Am.\ J. Math.\ 120, 117--128 (1998; Zbl 0938.58007)]. The current paper discusses what could be the reason for that different behaviour. Basically, the answer is that the calculations for equivariant Yang-Mills solutions behave like they had degree 2 singularities, while equivariant singularities to the harmonic map heat flow want to have degree 1. To support this statement, the authors prove two things: (1) The equivariant harmonic map heat flow does not blow up if the group action forces any possible singularity to have degree \(>1\). (2) The Yang-Mills heat equation, reduced by symmetry and modified in such a way that singularities would correspond to degree 1 singularities of the harmonic map flow, does blow up in finite time for suitable initial data.