If we go back to your earlier question on Frost diagrams (I've reproduced the key figure below), we see why simple molecular orbital theory or the "$4n+2$" rule suggests that benzene is aromatic while cyclobutadiene is antiaromatic.
Forming a planar, conjugated, 6-membered ring and placing 6 π-electrons in it creates a π-system that is energetically more stable than 3 conjugated C=C bonds (e.g. in hexa-1,3,5-triene). On the other hand, forming a planar, conjugated, 4-membered ring and placing 4 π-electrons in it creates a π-system that is energetically less stable than 2 conjugated C=C bonds (e.g. in butadiene). For these reasons we say that the first system, benzene, is "aromatic", while the second system, cyclobutadiene, is "antiaromatic". Other measurements and physical phenomenon such as reactivity, bond lengths, ring currents, etc. support these conclusions.
Free cyclobutadiene has been observed as a transient intermediate. Further theoretical analysis suggests that due to its lack of aromaticity it will distort from a square structure to a rectangular one with alternating single and double bonds (Jahn–Teller effect).
sources claim that this instability can be attributed to other factors such as ring and angle strain rather than antiaromaticity
It turns out that other systems involving the cyclobutadiene skeleton have been prepared and studied. Derivatives of both the cyclobutadiene dication ($4n+2$, $n=0$) and the cyclobutadiene dianion ($4n+2$, $n=1$) have been prepared and studied. Their stability is much less than that of benzene, but much more than that of cyclobutadiene. Placing two charges in these small rings results in extremely high coulombic repulsions in both the dianion and dication, and this may be a significant part of the explanation as to why they are less stable than say, benzene. The fact that they exist as planar structures, with appropriate ring currents and are more stable than cyclobutadiene suggests that the instability observed with cyclobutadiene is not due to ring strain and angle strain alone. Rather the aromatic stabilization - antiaromatic destabilization suggested by simple MO theory seems to be involved.
Olah, G. A.; Staral, J. S. Novel aromatic systems. 4. cyclobutadiene dications. J. Am. Chem. Soc. 1976, 98 (20), 6290–6304. DOI: 10.1021/ja00436a037.
Takanashi, K.; Inatomi, A.; Lee, V. Y.; Nakamoto, M.; Ichinohe, M.; Sekiguchi, A. Tetrakis(trimethylsilyl)cyclobutadiene dianion alkaline earth metal salts: new members of the 6π-electron aromatics family. Eur. J. Inorg. Chem. 2008, 2008 (11), 1752–1755. DOI: 10.1002/ejic.200800066.