Going back to late antiquity and early medieval times, there were problems, John Paul II admits. Iconoclasm becomes a factor:
Along this path there were troubled moments. Precisely on the issue of depicting the Christian mystery, there arose in the early centuries a bitter controversy known to history as “the iconoclast crisis”. Sacred images, which were already widely used in Christian devotion, became the object of violent contention. The Council held at Nicaea in 787, which decreed the legitimacy of images and their veneration, was a historic event not just for the faith but for culture itself.
So one question might be: does the fact of the incarnation give Christians freedom to move beyond our roots in Judaism, and the concerns of Islam about imaging God? JP2 says yes:
The decisive argument to which the Bishops appealed in order to settle the controversy was the mystery of the Incarnation: if the Son of God had come into the world of visible realities—his humanity building a bridge between the visible and the invisible— then, by analogy, a representation of the mystery could be used, within the logic of signs, as a sensory evocation of the mystery. The icon is venerated not for its own sake, but points beyond to the subject which it represents.(Cf. JOHN PAUL II, Apostolic Letter Duodecimum Saeculum (4 December 1987), 8-9: AAS 80 (1988), pp. 247-249)
Theoretically, icons point to the reality behind the image. Yet the danger of idolatry remains. Should Christians be more vigilant about this? Or have sufficient precautions been upheld. We will always have people who will take good things to extremes–that’s a human given.
Pope John Paul II’s Letter To Artists is available in its entirety online here.
Catholic Sensibility 
Well, the interesting thing is the varying tradition in depicting the Persons of the Holy Trinity. The depiction of Jesus, of course, evolved over the early centuries, both direct depiction and figurative (the depiction of the Lamb of God – and the ire of iconoclasts in relation to it – is one reason the litany about the Lamb of God found an enduring presence in the Roman ritual). The Holy Spirit coming in the visible form of a dove likewise allowed for figurative depiction in that manner. But the Father is a different matter: when the Father was depicted, if ever, it tended to be by oblique reference: light (in the case of stained glass, the light passing through could be taken as part of the depiction), a hand, an eye (remember, that eyes are an important component of depicting the highest orders of angels), an angel, et cet. IIRC, the Western practice of depicting the Father as the Ancient of Days as an old man didn’t become commonplace until after naturalism became well established in Western Christian sacred art, and there’s much to be said in favor of the older, more traditional approaches.