The error:

Cannot read property defaultValue of undefined

Is because the ref is not available so soon:

An important note about the ref registration timing: because the refs
themselves are created as a result of the render function, you cannot
access them on the initial render – they don’t exist yet! $refs is
also non-reactive, therefore you should not attempt to use it in
templates for data-binding.

And when you add it to the button‘s template, it tries to use it too soon.

The workaround would be to add a simple conditional:

<button type="submit" :disabled="!$refs.input || $refs.input.defaultValue == $refs.input.value">Submit</button>

But don’t be happy just yet.

The defaultValue won’t have the value you think

When using v-model, defaultValue will actually always be empty string ("") because Vue initially renders the <input> with an empty value.

To use a variable in the disabled button like you want, my suggestion is: use a mouted() logic to "save" the initial value and, in the button template, compare to it instead of defaultValue.

Demo below.

new Vue({
  el: '#app',
  data: {
    name: 'Hello Vue.js!'
  },
  mounted() {
    this.$refs.input.dataset.defVal = this.$refs.input.value;
  }
})

<script src="https://unpkg.com/vue"></script>

<div id="app">
  <p>{{ name }}</p>
  <input type="text" v-model="name" ref="input" />
  <button type="submit" :disabled="!$refs.input || $refs.input.dataset.defVal == $refs.input.value">Submit</button>
</div>

Alternative: Going Vue all the way

Of course, if it’s a possibilty, you should take advantage of Vue’s data-driven reactive nature, as tackling with the DOM is always tricky.

The solution would be to just create another variable and populate it on mounted():

new Vue({
  el: '#app',
  data: {
    name: 'Hello Vue.js!',
    defaultName: null
  },
  mounted() {
    this.defaultName = this.name;
  }
})

<script src="https://unpkg.com/vue"></script>

<div id="app">
  <p>{{ name }}</p>
  <input type="text" v-model="name"/>
  <button type="submit" :disabled="name == defaultName">Submit</button>
</div>

Or, if you can set both name and defaultName to the same initial value, the mounted() logic above wouldn’t even be necessary.