Both solutions should work, but it didn't. But after I transferred the entire solution over to the webhotel again, it now works also with reCAPTCHA (I didn't test mail server locally on my development machine, therefore probably these solutions have not been identical).

Now the entire solution is working, and it looks like this:

ContactFormSurfaceController.cs

using Neoweb.Models;
using Newtonsoft.Json;
using Newtonsoft.Json.Linq;
using System;
using System.Collections.Generic;
using System.Linq;
using System.Net;
using System.Net.Mail;
using System.Web;
using System.Web.Mvc;
using Umbraco.Web.Mvc;

namespace Neoweb.Controllers
{
    public class ContactFormSurfaceController : SurfaceController
    {
        // GET: ContactFormSurface
        public ActionResult Index()
        {

            if (TempData["reCaptcha"] == null)
            {
                ViewBag.MessageToUser = "Kontakt Neoweb";
            }

            return PartialView("ContactForm", new ContactFormViewModel());
        }

        [HttpPost]
        public ActionResult ContactFormSubmit(ContactFormViewModel model)
        {

            //Validate Google recaptcha here
            var response = Request["g-recaptcha-response"];
            string secretKey = "xxxxxxxxxxxxxx HIDED xxxxxxxxxxxxxx";
            var client = new WebClient();
            var result = client.DownloadString(string.Format("https://www.google.com/recaptcha/api/siteverify?secret={0}&response={1}", secretKey, response));
            var obj = JObject.Parse(result);
            var status = (bool)obj.SelectToken("success");
            TempData["reCaptcha"] = status;

            if (!ModelState.IsValid)
            {
                ViewBag.MessageToUser = "Alle felt må utfylles";
                return CurrentUmbracoPage();
            }

            if (!status)
            {
                // ViewBag.MessageToUser = status ? "reCaptcha validation success" : "reCaptcha validation failed";
                ViewBag.MessageToUser = "Kryss av for ikke robot";
                return CurrentUmbracoPage();
            }
            else
            {
                // Send email
                MailMessage message = new MailMessage();
                message.To.Add("[email protected]");
                message.Subject = "New message";
                message.From = new System.Net.Mail.MailAddress(model.Email, model.Name);
                message.Body = model.Message;
                SmtpClient smtp = new SmtpClient();
                smtp.Send(message);

                TempData["success"] = true;

                return RedirectToCurrentUmbracoPage();
            }    
        }
    }
}

ContactForm.cshtml

@model Neoweb.Models.ContactFormViewModel
@if (TempData["success"] == null)
{
    <div class="well-sm contact-form">
        @using (Html.BeginUmbracoForm<Neoweb.Controllers.ContactFormSurfaceController>("ContactFormSubmit"))
        {
            <fieldset>

                <legend class="">@ViewBag.MessageToUser</legend>

                <div class="form-group">
                    <span class="fa fa-user bigicon"></span>
                    @Html.TextBoxFor(m => m.Name, new { @class = "form-control", placeholder = "Fullt navn" })
                    @Html.ValidationMessageFor(m => m.Name, "Navn er påkrevd.")
                </div>
                <div class="form-group">
                    <span class="fa fa-envelope-o bigicon"></span>
                    @Html.TextBoxFor(m => m.Email, new { @class = "form-control", placeholder = "E-post" })
                    @Html.ValidationMessageFor(m => m.Email, "E-post adresse er påkrevd.")
                </div>
                <div class="form-group">
                    <span class="fa fa-pencil-square-o bigicon"></span>
                    @Html.TextAreaFor(m => m.Message, new { @class = "form-control", placeholder = "Melding", rows = "7" })
                    @Html.ValidationMessageFor(m => m.Message, "Meldingstekst er påkrevd.")
                </div>
                <div class="g-recaptcha" data-sitekey="xxxxxxxxxxxxxx HIDED xxxxxxxxxxxxxx"></div>
                <div class="form-group">
                    <button type="submit" class="btn btn-primary">Send melding</button>
                </div>
            </fieldset>
        }
    </div>
}
else
{
    <div class="well-sm">
        <p>Takk, du hører snart fra oss</p>
    </div>
}

ContactFormViewModel.cs (Model):

using Newtonsoft.Json;
using System;
using System.Collections.Generic;
using System.ComponentModel.DataAnnotations;
using System.Linq;
using System.Web;

namespace Neoweb.Models
{
    public class ContactFormViewModel
    {
        [Required]
        public string Name { get; set; }
        [Required]
        [EmailAddress]
        public string Email { get; set; }
        [Required]
        public string Message { get; set; }
    }
}

ContactForm.cshtml (Partial View Macro File):

@inherits Umbraco.Web.Macros.PartialViewMacroPage
@Html.Action("Index", "ContactFormSurface")

Everything is now OK, but ideas if the solution can be improved further will be appreciated