Am I correct in thinking your sister site is still using the older 2.0.11 release? The basic problem may be that on the site with the problem the widget structure has changed (in line with the update to 3.1) but the corresponding CSS rules you/your developer might have added have not been updated.
It doesn’t actually look too bad to me, but you can target it and tweak it to make it look how you want with rules like this (this example gives it a white background, as on the sister site):
.widget-wrap .tribe-mini-calendar-grid-wrapper { background: #fff; }
Which you might add to a custom tribe-events.css stylesheet. There are details about implementing this and more in our Themer’s Guide – does that help at all?