How does the Filioque affect Reformed Protestant theology? In the past, extremely influential Eastern theologians (Photius the Great, Mark of Ephesus, etc.) have rejected the Filioque on important theological grounds.

Likewise, the Latin Catholic church insists on the insertion of "and the Son" into the Nicene creed.

What is the weight of importance for the Filioque   in Reformed theology...or is it somewhat of a non-factor in the grand scheme?
The Filioque is considered to be of extremely significant importance in Reformed Theology. Even in Reformed-flavored Protestantism (such as that found at The Gospel Coalition), it has implications for Scripture's authority, as well as preaching and discipleship. These implications include the authority of Scripture and the preaching and discipleship of the church: the Scriptures must be our final authority for deciding matters of faith and practice, rather than the councils of fallible men (which are useful insofar as they are in agreement with the Word of God) ; for preaching and discipleship--when Christians are commanded to live holy lives, they are enabled to do so by the Spirit of Christ in them.

For it's significance in specifically Reformed thinking, Richard Muller writes in his Post Reformation Reformed Dogmatics, 


  The Reformed exegetes, moreover, understood the issue to be one of exegesis, not merely an issue of the form of the Niceno-Constantinopolitan Creed, and found the biblical text to be entirely of one accord in favor of double procession.


Which is further confirmation of the Reformed view that Scripture is the ultimate authority for questions of faith and practice.

In a lecture delivered at Mid-America Reformed Seminary, Nov. 10, 1999, Robert Letham says,


  According to the West the Eastern repudiation of the filioque 
  leaves no clear relation between the Son and the Holy Spirit. This is 
  in odd contrast to the patristic teaching of perichoresis, whereby the 
  persons of the Trinity indwell and interpenetrate one another. The 
  West holds that this exhibits subordinationist tendencies reaching 
  back as far as Origen, for in the East the Son and the Holy Spirit are 
  commonly said to derive their deity from the Father. In contrast, 
  the filioque affirms the intimate relation between the Son and the 
  Spirit, and thus between the Word and Spirit. The East’s position, 
  the West claims, has led to a gulf between theology and piety. 
  Speculative theology, grounded on the Logos, has been separated 
  from worship, mediated by the Holy Spirit. Thus Eastern piety, so 
  Western observers like Bavinck claim, is unduly dominated by 
  mysticism. 14


Although many historic Reformed theologians would disagree with him, Letham considers this argument to actually be flawed. So he offers what he considers a stronger argument:


  A third objection, however, carries much greater weight. Following John of Damascus, the East tends to consider that the essence of God is unknowable, only God’s energies or operations being revealed, the things around him (“all that we can affirm concerning God does not shew forth God’s nature, but only the qualities of his nature”-- τα περι την φυσιν). 17 This dichotomy is used to offset some of the Biblical evidence for the joint and coordinate involvement of the Son in relation to the Holy Spirit. As a sympathetic critic like T.F. Torrance argues, it drives a wedge between the inner life of God and his saving activity in history, ruling out any real access to knowing God in himself.18 It also departs from earlier Greek patristic thought, which rejected this distinction.19 Besides opening a yawning chasm between the economic and the ontological, the tendency seems to be towards a quarternity rather than a trinity—the unknowable divine essence plus the three revealed persons.


It's worth noting that Letham ultimately comes down somewhat in the middle on the Filioque, and suggests in his lecture that both West and East could come to agreement on the topic. Since Letham's work is relatively recent (1999), I'm not sure how widely this is accepted.

In this article Bojidar Marinov (a Bulgarian Reformed missionary leader) writes about the historical and cultural implications of the Filioque for the East and West. He concludes that the Filioque is the prime reason for the theological and cultural differences between the East and West.


  Faith has consequences. What started as an innocent difference in wording, by one word only, actually led over a long period of time to a huge and not so innocent difference in faith, then ideology, then social practice, then culture. The changes were not immediate, but even as early as the 9th and the 10th centuries it was visible. The Western Church was compiling the Canon Law; the Eastern Church was compiling the Lives of the Saints. The Western Church was fighting kings and emperors over the validity of the old royal/pagan laws; the Eastern Church was writing treatises on the emperors as divine legislators. The Western Church was developing the idea of practical imitation of Christ; the Eastern Church was developing the idea of the mystical imitation of Christ. Christ’s place in the representative work of the Spirit made the difference. The filioque made the difference. 


So for the historical Reformed position, the Filioque not only develops from the text of Scripture and impacts how one understands who God is, but it also uniquely affects the history and piety of the church. 

This is particularly evident in understanding how it was good for the church that Jesus went away physically. In John 16:7, ESV, Jesus said, "Nevertheless, I tell you the truth: it is to your advantage that I go away, for if I do not go away, the Helper will not come to you. But if I go, I will send him to you." Though the disciples were disturbed that Jesus said he was going away, they didn't know what to make of his words. Jesus can't have been referring to his second coming because in John 14:19 ,using nearly identical language he says that "yet a little while and the world will see me no more, but you will see me." So the seeing wasn't going to be that of the world seeing him return whereby they would fall down in fear before him, nor was it going to be the world seeing him which resulted in them hating and killing him. Rather, the seeing was going to be the church having Jesus' real spiritual presence by His Spirit. Jesus rose and ascended bodily, but he sent the Spirit to the church to fulfill his promise that he would come again to them after he had left. 

The Spirit reveals Christ. In Reformed theology, this is seen as the chief work of the Spirit. People sometimes divorce the work of the Spirit from that of making Christ known, but the Spirit hasn't spoken of himself as having a separate work besides that of making Christ known. This illuminating work of the Holy Spirit is what makes Christ known as more and more beautiful to the believer. If the Holy Spirit only proceeds from the Father, then He can provide access to the Father apart from Christ. This goes against the biblical teaching that our access to the Father is through the Son, and by the Spirit. The Spirit reveals Christ to us, and through Christ we have access to the Father. The filioque affects the very core of the Christian life--union and communion with the Triune God.

Addendum, if anyone has a subscription to WTJ (perhaps also available through your theological library or even interlibrary loan), they could see more than the first page of this article which is specifically on the importance of the filioque for Reformed theology.