boost converter vs transformer
That is it exactly. Transformers only work on AC. Put DC into a transformer, it will basically act as a short circuit.

A boost converter (actually any switching regulator) is actually just a way to use DC in a transformer by converting it to AC. In a simplified view, a boost converter consists of three parts: an inverter to turn DC into AC (square wave, not sine wave); a transformer; and a rectifier/filter to turn it back into DC.

The additional stuff I'm ignoring include a feedback system to vary the duty cycle of the square wave to regulate the output voltage and the fact that because the square wave is much higher frequency than the mains, the transformer can be much smaller.

Edit: it was pointed out that I was describing a flyback converter, not a boost converter. A boost converter instead uses an inductor to store and release energy as switched current (effectively AC) is fed to it. However, my point stands: in order to increase voltage from DC, you must turn it into AC first to allow something to transform it into a higher voltage.