Intelligent Design is a very broad description. It covers a range of ideas that do not necessarily include the Bible. Even the term "Creationism" doesn't always cover the Christian view. A Muslim can just as easily be called a Creationist. Biblical Creationism is a very specific form of Intelligent Design. There are aspects of the belief that are unique to it.
One unique aspect of Biblical Creationism is the Fall of man. In Genesis 3, an event is described in which Adam and Eve, the first two humans, rebelled against God's commands. God then cursed the entire creation (Romans 8:22), resulting in a number of changes. Some of those changes are specifically mentioned, such as death, the necessity of work, a woman's pain during childbirth, and the creation of thorns and thistles. Other results are not directly mentioned, but they are alluded to. God tells Satan that he is cursed "above all livestock" and "more than every beast of the field" (Genesis 3:14), but He does not directly describe what, exactly, the livestock and beasts of the field have been cursed with.
One thing we do see is that whereas all animals were initially vegetarian (Genesis 1:29-30), this is obviously not so after the fall. Indeed, when the same command to eat only plants is revoked for man after the flood (Genesis 9:3), we never see this being revoked for animals, which may suggest they had already descended into carnivory. Further evidence of this can be seen in the claim that "all flesh" had corrupted its way (Genesis 6:12).
Further changes were made to the earth by way of the flood. Peter tells us that the world which then was perished (2 Peter 3:6). It is even likely that there was once a giant supercontinent (Genesis 1:9) that may well have been broken up during the catastrophe of the flood. The Garden of Eden is certainly long gone, impossible to locate. All this to say the world we observe today is not the same world God created, nor is it the same world that Noah knew before the flood. Everything is different now.
This fact disarms a number of objections presented by Evolutionists against Creationism. Effectively, any argument that follows the structure of "why did God create..." and is followed by an example of something horrible, like disease, violence, or chaos, can be answered with "it wasn't like that when God created it", and in some cases can also be followed up with "but it is necessary in a fallen world because...". Similarly, the argument that things go extinct cannot be used as an argument against Biblical Creation because extinction is to be expected. There's even some evidence that God may have directly wiped out certain species Himself (e.g. Isaiah 27:1; Job 40:19).
Natural selection is also a perfectly valid design feature in a fallen world. God knew that there would be drastic changes in the world from the fall to the flood to the modern day, and so natural selection, while sometimes used as an argument against a God of love due to its brutality, is actually a critical feature that prevented extinction level events so brutal, the modern world would be lifeless and barren.
To illustrate just how flawed these arguments are, consider the cactus in the above image. It's not all one cactus. It's actually two cacti grafted together. The orangey red top is a species called red moon, which has been grafted onto a rootstock. This grafting is necessary, because because the red moon cannot produce chlorophyll, thus it cannot photosynthesise, effectively dooming the red moon to a parasitic life, as it would not be able to survive independently.
Lacking this knowledge, an Evolutionist might ask "why did God create a cactus that can't even photosynthesise without being grafted onto a rootstock?" But this is the wrong question to ask, because God did not create a cactus that cannot photosynthesise. Rather, the red moon is a mutant version of a plant that can photosynthesise. It has lost its God-given ability to photosynthesise via mutation. Speaking as a man who owns this very plant, I am glad it exists, because I find it beautiful and fascinating.
It is this ignorance of Biblical Creationism that gives many people cause to reject it. If more Evolutionists understood Biblical Creationism, a number of arguments against it would swiftly dissipate, and Evolution would cease to be the "only" explanation for these things. When discussing origins, the Fall must be taken into account. Those who fail to do so are either unqualified, or dishonest.