This is a rather brief answer - but I think it will be helpful.

Supernatural and Miraculous can be used interchangeably, and it is appropriate to use them in the same contexts at certain points. However, I think there is an actual distinction to be made.

Supernatural - a realm outside of the natural and material world. Generally thought to contain angels, spirits, demons, God/gods, and other non-physical beings. This could also include the paranormal, depending on the context.

Miraculous - generally, a reference to events that contain miracles. While these events can be natural (ex: abnormal recovery from cancer for natural reasons), it is better to be used (in the religious context) as an event that violates the laws of nature. (Ex: people crawling up walls while demon possessed and violating the laws of gravity). The miraculous is almost an “extension” of the supernatural, as miraculous events flow from the supernatural.

Hope this helps :)

I just realized you asked for a biblical basis - but I don’t see this as really a “Bible issue”, more of a definitional question.

The "Biblical basis" may be as mundane here as a matter of grammar. The Bible uses "miracle" as a noun.

The word "supernatural" is usually an adjective, and is only made a noun by the addition of the definite article, as in "the supernatural"--a form which does not fit in any of the places in the Bible where "miracle" is used, and, furthermore, this form is unattested as of Webster's 1828 Dictionary.

We would not say, for example, "When Pharaoh shall speak unto you, saying, Show {the supernatural} (a miracle) for you: . . ." (Exodus 7:9). Nor could we ever use "supernatural" as a plural noun, such as in "And his {supernaturals} (miracles), and his acts, which he did in the midst of Egypt unto Pharaoh the king of Egypt, and unto all his land;" (Deuteronomy 11:3).

Essentially, the word "miracle," in both singular and plural forms, is the only nominative equivalent in English that translators could have used when selecting between "miracle" and "supernatural." Beyond those options, they could choose among "signs," "wonders," "marvels," etc., and sometimes did use one of those forms; but "supernatural" is simply not a noun.

A word that comes closer to the concept of supernatural is "elohim." This is explained by Hebrew scholars as actually being a reference to superior and/or supernatural beings.

The word "supernatural" would parallel "miraculous" in usage, and the word "miraculous" is unattested in the KJV. Evidently, the adjective form is not used. However, the word "supernatural" is linked to the word "miracle" in Webster's 1828 Dictionary.

OP: What is the difference between "miraculous" and "supernatural"?

Only God can do the miraculous, which would also be supernatural.

Satan and demons can operate in the supernatural and perform supernatural acts, but can never do miracles.

For example, Pharaoh's magicians could turn their sticks into snakes, as could Moses. Pharaoh, however, asked for a miracle.

So Moses and Aaron went to Pharaoh and did just as the LORD commanded. Aaron threw his staff down in front of Pharaoh and his officials, and it became a snake. Pharaoh then summoned wise men and sorcerers, and the Egyptian magicians also did the same things by their secret arts: Each one threw down his staff and it became a snake. But Aaron’s staff swallowed up their staffs. Exo 7:10-12

Both operated in the supernatural arena, but only Aaron's staff brought about a miracle.

Recall also what Christ said about miracles.

[after healing a demon-possessed man] But when the Pharisees heard this, they said, “It is only by Beelzebul, the prince of demons, that this fellow drives out demons.”Jesus knew their thoughts and said to them, “Every kingdom divided against itself will be ruined, and every city or household divided against itself will not stand. If Satan drives out Satan, he is divided against himself. How then can his kingdom stand? And if I drive out demons by Beelzebul, by whom do your people drive them out? So then, they will be your judges. But if it is by the Spirit of God that I drive out demons, then the kingdom of God has come upon you. Mt 12:24-28

Satan operates in the supernatural, Christ operates in the supernatural also, but is the only One who brings about miracles. In this example, Satan possessed the man, but Christ drove it out, causing a miracle.

In short, the supernatural is the arena, but miracles are solely from God.

Supernatural

Supernatural means what exceeds nature. It can be divided as follows (Garrigou-Lagrange, O.P., The Sense of Mystery p. 255):

^{"Miracle" is on the bottom right corner of these diagrams.}

A miracle is something supernatural "from efficiency" (ex parte efficientis)—i.e., the efficient cause of a miracle (God*) is supernatural.

^{*Only God can work miracles: "…qui facit mirabilia solus" (Ps. 71:18)}

A more detailed version of this diagram is given on Woodbury, S.M., The Supernatural and Grace p. 11:

Types of miracles

St. Thomas Aquinas distinguishes the three types of miracles, miracles being "works that are sometimes done by God outside the usual order assigned to things", in Summa Contra Gentiles, lib. 3 cap. 101 (from highest to lowest):

1. "those events in which something is done by God which nature never could do."
2. "those events in which God does something which nature can do, but not in this order"
3. "when God does what is usually done by the working of nature, but without the operation of the principles of nature."

^(source)

What is the difference between "miraculous" and "supernatural"?

The miraculous or miracle is always supernatural. Yet not all supernatural happenings are miraculous. Something may be supernatural and yet not miraculous. There are nuances that must be somehow understood.

The miraculous extends beyond the realm of the natural. Physical healing brought about by natural means does not constitute a miraculous event.

A miracle in essence is an effect or extraordinary event in the physical world that surpasses all known human or natural powers and is ascribed to a supernatural cause.

St. Thomas affirms that true miracles are brought about by God alone.

True miracles cannot be wrought save by the power of God, because God works them for man's benefit, and this in two ways: in one way for the confirmation of truth declared, in another way in proof of a person's holiness, which God desires to propose as an example of virtue. On the first way miracles can be wrought by any one who preaches the true faith and calls upon Christ's name, as even the wicked do sometimes. On this way even the wicked can work miracles. Hence Jerome commenting on Matthew 7:22, "Have not we prophesied in Thy name?" says: "Sometimes prophesying, the working of miracles, and the casting out of demons are accorded not to the merit of those who do these things, but to the invoking of Christ's name, that men may honor God, by invoking Whom such great miracles are wrought."

In the second way miracles are not wrought except by the saints, since it is in proof of their holiness that miracles are wrought during their lifetime or after death, either by themselves or by others. For we read (Acts 19:11-12) that "God wrought by the hand of Paul . . . miracles" and "even there were brought from his body to the sick, handkerchiefs . . . and the diseases departed from them." On this way indeed there is nothing to prevent a sinner from working miracles by invoking a saint; but the miracle is ascribed not to him, but to the one in proof of whose holiness such things are done. - Question 178. The grace of miracles

The supernatural interventions of Satan and the evil spirits are not genuine miraculous events. Demons cannot work miracles as such, since this belongs solely to God as something done outside the order of created nature. They can perform what appears to be miraculous in that it exceeds human power and experience.

If Satan could impact the weather, it would only be by God’s permission, and restrained, as in the case of Job. Satan was allowed by God to torment Job in order to test him, and this included “the fire of God” (probably lightning) which "fell from the sky and burned up the sheep and the servants" (Job 1:16). This was followed by a "mighty wind" (possibly a tornado) that destroyed his eldest son’s home and killed Job’s children (verses 18-19). So if the fire from heaven and the tornado were somehow caused by Satan, they were still under the ultimate control of God for His purposes.

Satan is seen as the “prince of the power of the air” in Ephesians 2:2 and the “god of this world” in 2 Corinthians 4:4 as evidence for Satan having control over weather. An examination of Scripture reveals that whatever influence Satan and his demon angels have over the weather is restricted by God’s ultimate sovereignty.

Demoniac supernatural influences are thus to be seen as false miracles and are not miraculous.

Angels cannot, of themselves, work miracles. A miracle, by definition, is a work proper to God alone. Of course, angels can serve, even as holy men may serve, as ministers or instruments in the performing of miracles. Angels, good or bad, can do wonderful things, but only such as lie within the power of angelic nature, and a miracle surpasses the powers of all created natures.

Holy Angels or Messengers aid and enlighten men. This is seen in Scriptures at the Annunciation Luke 1:26-38, or when an Angel spoke to Zachary, the father of St. John Baptist Luke 1:5-25. Bad Angels influenced people to do what is sinful, such as when Eve was tempted in the garden by Satan. These are all supernatural events, but they are not miracles in the true sense of the word.

C.S. Lewis viewed consideration of the miraculous, that is to say one's perception of a perceived or recorded event within nature, to be directly informed by one's worldview; i.e. naturalist or super-naturalist:

“…the question whether miracles occur can never be answered simply by experience …. This book is intended as a preliminary to historical inquiry…. Those who assume that miracles cannot happen are merely wasting their time by looking into the texts: we know in advance what results they will find for they have begun by begging the question” - Miracles-a preliminary study

Whereas naturalism claims that the natural world is all there is and there are no supernatural beings, super-naturalism claims that there are beings which are "other than" or outside of nature and that these being can and sometimes do enter a cause and effect relationship with the natural world. Thus, in Lewis' view, it is the predisposition of the observer that makes an event appear miraculous and not the event itself and it is genuine conversion that allows one to perceive the miraculous by dint of a worldview change:

He that hath received his testimony hath set to his seal that God is true. - John 3:33

This is not to say that Lewis considered miracles as figments of the imagination. An occurrence within nature has, ontologically, a natural or supernatural cause (Lewis argues it is often both). Therefore we do not "make" miracles by our preconceptions but we dismiss or accept them accordingly.

A miracle, then, (the miraculous) is an event brought about in nature by, at least in part, a supernatural cause which is outside of nature. Biblically speaking, super-naturalism is the default assumption and the very existence of the natural is miraculous:

In the beginning God ... Genesis 1:1