Unfortunately the colorbar affects aspect ratio. If you turn it off, you should get same size figures.
The downside is that there is no good way (I know of) to plot the colorbar separately. A hackish way is:
using Images, FFTW, Plots
gr()
img = load("test.png")
F = fftshift(fft(Float64.(Gray.(img))))
tf = log.(abs.(F.*F)).+1
imgdisp = plot(img, aspect_ratio=1, xlims=(1,size(img,2)), ylims=(1,size(img,1)))
fspec = heatmap(tf, aspect_ratio=1,colorbar=nothing, xlims=(1,size(tf,2)), ylims=(1,size(tf,1)), clims=extrema(tf) )
cbar = heatmap(minimum(tf).*ones(1,1), aspect_ratio=10000, left_margin=-5Plots.mm, right_margin=5Plots.mm, framestyle=:none, clims=extrema(tf) )
plot(imgdisp, fspec, cbar, layout = grid(1, 3, widths=[0.4 ,0.4, 0.2]) )
