Help with a Web Server Logic error

You need to decouple receiving input from displaying your patterns. Read this tutorial: several things at the same time

And then structure your code like a state machine. Updating/running your patterns should be moved outside your if(client) {} condition since you want to update your patterns every time though loop.

#include <WiFi.h>
#include <Adafruit_NeoPixel.h>
#ifdef __AVR__
#include <avr/power.h>
#endif

#define PIN        13
#define NUM_LEDS 300


Adafruit_NeoPixel strip(NUM_LEDS, PIN, NEO_GRB + NEO_KHZ800);

#define DELAYVAL 500

void showStrip() {
#ifdef ADAFRUIT_NEOPIXEL_H
  // neopixel
  strip.show();
#endif
#ifndef ADAFRUIT_NEOPIXEL_H
  // fastlED
  FastLED.show();
#endif
}

void setPixel(int Pixel, byte red, byte green, byte blue) {
#ifdef ADAFRUIT_NEOPIXEL_H
  // NeoPixel
  strip.setPixelColor(Pixel, strip.Color(red, green, blue));
#endif
#ifndef ADAFRUIT_NEOPIXEL_H
  // FastLED
  leds[Pixel].r = red;
  leds[Pixel].g = green;
  leds[Pixel].b = blue;
#endif
}

void setAll(byte red, byte green, byte blue) {
  for (int i = 0; i < NUM_LEDS; i++ ) {
    setPixel(i, red, green, blue);
  }
  showStrip();
}


uint16_t rainbow_j;

void rainbowCycle(int SpeedDelay) {
  byte *c;
  uint16_t j = rainbow_j;

  for (int i = 0; i < NUM_LEDS; i++) {
    c = Wheel(((i * 256 / NUM_LEDS) + j) & 255);
    setPixel(i, *c, *(c + 1), *(c + 2));
  }
  strip.setBrightness(255);
  showStrip();
  delay(SpeedDelay);
  rainbow_j++;
  if ( rainbow_j >= 256 * 5 ) rainbow_j = 0;
}

byte * Wheel(byte WheelPos) {
  static byte c[3];

  if (WheelPos < 85) {
    c[0] = WheelPos * 3;
    c[1] = 255 - WheelPos * 3;
    c[2] = 0;
  } else if (WheelPos < 170) {
    WheelPos -= 85;
    c[0] = 255 - WheelPos * 3;
    c[1] = 0;
    c[2] = WheelPos * 3;
  } else {
    WheelPos -= 170;
    c[0] = 0;
    c[1] = WheelPos * 3;
    c[2] = 255 - WheelPos * 3;
  }

  return c;
}


const char* ssid     = "BT-TGAHZR";
const char* password = "beqxyVRt6cGng7";

WiFiServer server(80);

void setup()
{
  Serial.begin(115200);
  pinMode(5, OUTPUT);

  delay(10);

  Serial.println();
  Serial.println();
  Serial.print("Connecting to ");
  Serial.println(ssid);

  WiFi.begin(ssid, password);

  while (WiFi.status() != WL_CONNECTED) {
    delay(500);
    Serial.print(".");
  }

  Serial.println("");
  Serial.println("WiFi connected.");
  Serial.println("IP address: ");
  Serial.println(WiFi.localIP());

  server.begin();
  strip.begin();
}


enum { NONE, RAINBOW };

int pattern = NONE;

void loop() {

  switch ( pattern  ) {
    case NONE:
      // nothing to do
      break;

    case RAINBOW:
      rainbowCycle(5);
      break;
  }
  
  WiFiClient client = server.available();

  if (client) {
    Serial.println("New Client.");
    String currentLine = "";
    while (client.connected()) {
      if (client.available()) {
        char c = client.read();
        Serial.write(c);
        if (c == '\n') {
          if (currentLine.length() == 0) {
            client.println("HTTP/1.1 200 OK");
            client.println("Content-type:text/html");
            client.println();
            client.print("Click <a href=\"/N\">here</a> to Neophase.<br>");
            client.print("Click <a href=\"/R\">here</a> to Running Lights.<br>");
            client.print("Click <a href=\"/RC\">here</a> to Rainbow Cycle.<br>");
            client.print("Click <a href=\"/L\">here</a> to turn off.<br>");
            client.println();
            break;
          } else {
            currentLine = "";
          }
        } else if (c != '\r') {
          currentLine += c;
        }
      }
      if ( currentLine.endsWith("GET /RC")) {
        pattern = RAINBOW;
        rainbow_j = 0;
      }
      if (currentLine.endsWith("GET /L")) {
        pattern = NONE;
        setAll(0, 0, 0);
      }
    }
  }
  client.stop();
  Serial.println("Client Disconnected.");
}