Basic arithmetic calculator in JavaScript
I would go with the second one. Not because eval here is insecure -- you're validating the arguments -- but because it's overkill. eval is designed to parse and run any JavaScript code, so it is slow (compared to normal JavaScript code); engines do a lot of optimisation before running code, but they can't optimise if the code is in a dynamic string which changes at runtime. It probably won't maker much difference for you here, but if you're building a larger app, blocking JavaScript can be noticeable.

Here are the changes I would make to calculate2:

'use strict'; // Strict mode === good

function calculate2(num1, num2, operator) {
  // Number.parseFloat is new in ES6
  // (it's identical to window.parseFloat, so you can use either)
  num1 = Number.parseFloat(num1);
  num2 = Number.parseFloat(num2);
  operator = operator.trim();

  // Avoid `var`. ES6 brings `let` and `const`, which:
  //   - are block scoped
  //   - work with destructuring
  //   - prevent redeclaration in the same scope
  //   - aren't hoisted
  // Note that `const` doesn't freeze object properties
  // If you need frozen properties, use Object.freeze
  const calculations = {
    // Shorter names, it's clear from the context what they do.
    '+' : (a, b) =&gt; a + b,
    '-' : (a, b) =&gt; a - b,
    '*' : (a, b) =&gt; a * b,
    '/' : (a, b) =&gt; a / b, // You can use trailing commas in object and array literals if you want.
  }; // If you use semicolons, use them all the time.

  // Detailed error messages.
  // 
  // Number.isNan is also new in ES6,
  // however it's not quite the same as window.isNan.
  // Here it won't make a difference which you use.
  // 
  // `throw new Error` rather than `throw Error`
  // as you're using the Error constructor.
  if (Number.isNan(num1)) throw new Error(
    `Expected first argument to calculate2 to be a String valid for Number.parseFloat but got ${num1}.`);
  if (Number.isNan(num2)) throw new Error(
    `Expected second argument to calculate2 to be a String valid for Number.parseFloat but got ${num2}.`);

  // Don't bother maintaining a separate regex of valid operators
  // Just check the calculations object
  if (!calculations[operator]) throw new Error(
    `Expected third argument to calculate2 to be one of ${Object.keys(calculations).join(' ')} but got ${operator}.`);

  return calculations[operator](num1, num2);
}