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Ampère, a half century before Maxwell, theorized that magnetism was caused by electrical currents. So, why is Maxwell and not Ampère credited for unifying electricity and magnetism?
(cf. the question "Why is one of Maxwell's equations named after Ampère?")
On my opinion, the following citation from Poincare answers this question:
“At the time, when Maxwell initiated his studies, the laws of electrodynamics adopted before him explained all known phenomena. He started his work not because some new experiment limited the importance of these laws. But, considering them from a new standpoint, Maxwell noticed that the equations became more symmetric, when a certain term was introduced into them, although, on the other hand, this term was too small to give rise to phenomena, that could be estimated by the previous methods. A priori ideas of Maxwell are known to have waited for their experimental confirmation for twenty years; if you prefer another expression, — Maxwell anticipated the experiment by twenty years". (H. Poincare, The value of Science).
Maxwell's contribution was introduction of the so-called "displacement current". Then his equations predicted the existence of electromagnetic waves, the thing experimentally confirmed by Hertz.
There are two reasons:
As explained by Alexandre Eremenko, the laws of electrodynamics before Maxwell (like Ampere's force law, Neumann's law of induction, Weber's law, etc.) explained all the known phenomena in an action at a distance manner. When action at a distance theories fell out of favour, the pioneers of these theories were mostly discredited. Not to mention the fact that these pioneers had their name attached to some part of field theory. For eg $-$ Ampere's circuital law, Gauss law, etc.
In the later accepted field theory, the pioneer was James Clerk Maxwell (Faraday did initiated field theory but it was Maxwell who mathematically expressed them). Not only he expressed them mathematically but also symmetrically.
From these two reasons it would be obvious that it was only Maxwell who would be eligible for being credited as "unifier of electricity and magnetism"
Here is a quote from Maxwell's treatise regarding field theory vs action at a distance theory:
In a philosophical point of view, moreover, it is exceedingly important that two methods should be compared, both of which have succeeded in explaining the principal electromagnetic phenomena, and both of which have attempted to explain the propagation of light as an electromagnetic phenomenon, and have actually calculated its velocity, while at the same time the fundamental conceptions of what actually takes place, as well as most of the secondary conceptions of the quantities concerned, are radically different. I have therefore taken the part of an advocate rather than that of a judge, and have rather exemplified one method than attempted to give an impartial description of both. I have no doubt that the method which I have called the German one (action at a distance) will also find its supporters, and will be expounded with a skill worthy of its ingenuity.
