[Vuejs]-Vee-validate different text-field values

Have a read of the documentation for cross field validation. Essentially the steps are

1. Wrap the fields within the same ValidationObserver component.
2. Give each field a name property
3. Reference the target field name or vid value in the rules of the other.

For your code:

<template>
  <v-app>
    <ValidationObserver ref="form" v-slot="{ invalid, validated, passes, validate }">
      <form>
        <ValidationProvider rules="required|min:6" v-slot="{ errors, valid }" name='password'>
          <v-text-field
            v-model="current_password"
            label="Current Password"
            type="password"
            :success="valid"
            :error-messages="errors"
          />
        </ValidationProvider>
        <ValidationProvider rules="required|distinct:@password" v-slot="{ errors, valid }" name='new password'>
          <v-text-field
            v-model="new_password"
            label="New Password"
            type="password"
            :success="valid"
            :error-messages="errors"
          />
        </ValidationProvider>
      </form>
    </ValidationObserver>
  </v-app>
</template>

<script>
import { ValidationObserver, ValidationProvider, Extend } from "vee-validate";

extend('distinct', {
  params: ['target'],
  validate(value, { target }) {
    return value !== target;
  },
  message: 'Fields can not be the same'
});

export default {
  data() {
    return {
      current_password: "",
      new_password: "",
    };
  },
  components: {
    ValidationObserver,
    ValidationProvider
  },
};
</script>