example stopwatch solution (taking catlikecoding's clock tutorial further)

Clock.cs

using System;
using UnityEngine;
using UnityEngine.UI;

public class Clock : MonoBehaviour {

	const float
		degreesPerHour = 30f,
		degreesPerMinute = 6f,
		degreesPerSecond = 6f;

	TimeSpan stopwatchStartCont;
	DateTime stopwatchStartDisc;

	public bool stopwatch = false;
	public bool stopwatchRunning = false;
	public Text digitalText;
	public Transform hoursTransform, minutesTransform, secondsTransform;
	public bool continuous;

	void Update () {
		HandleInput();
		if (continuous) {
			UpdateContinuous();
		}
		else {
			UpdateDiscrete();
		}
	}

	void HandleInput ()
	{
		if (Input.GetKeyDown("space"))
		{
			if (!stopwatch && !stopwatchRunning)
			{
				stopwatch = true;
				ResetStopwatch();
			}
			else if (stopwatch && !stopwatchRunning)
			{
				stopwatchRunning = true;
				stopwatchStartCont = DateTime.Now.TimeOfDay;
				stopwatchStartDisc = DateTime.Now;
			}
			else if (stopwatch && stopwatchRunning)
			{
				stopwatchRunning = false;
				ResetStopwatch();
			}
		}
		if (Input.GetKeyDown("escape"))
		{
			stopwatch = false;
			stopwatchRunning = false;
		}
		if (Input.GetKeyDown("c"))
		{
			continuous = !continuous;
		}
	}

	void ResetStopwatch ()
	{
		UpdateDigital(0, 0, 0);
		hoursTransform.localRotation =
			Quaternion.Euler(0f, 0f, 0f);
		minutesTransform.localRotation =
			Quaternion.Euler(0f, 0f, 0f);
		secondsTransform.localRotation =
			Quaternion.Euler(0f, 0f, 0f);
	}

	void UpdateDigital (int hour, int minute, int second)
	{
		digitalText.text = hour.ToString().PadLeft(2, '0') + ":" + minute.ToString().PadLeft(2, '0') + ":" + second.ToString().PadLeft(2, '0');

	}

	void UpdateContinuous () {
		TimeSpan time = DateTime.Now.TimeOfDay;
		if (!stopwatch)
		{
			float hour = (float)time.TotalHours;
			float minute = (float)time.TotalMinutes;
			float second = (float)time.TotalSeconds;

			hoursTransform.localRotation =
				Quaternion.Euler(0f, hour * degreesPerHour, 0f);
			minutesTransform.localRotation =
				Quaternion.Euler(0f, minute * degreesPerMinute, 0f);
			secondsTransform.localRotation =
				Quaternion.Euler(0f, second * degreesPerSecond, 0f);

			UpdateDigital((int)time.Hours, (int)time.Minutes, (int)time.Seconds);
		}
		if (stopwatchRunning)
		{
			TimeSpan digitalTime = time - stopwatchStartCont;
			float stopHour = (float)time.TotalHours - (float)stopwatchStartCont.TotalHours;
			float stopMinute = (float)time.TotalMinutes - (float)stopwatchStartCont.TotalMinutes;
			float stopSecond = (float)time.TotalSeconds - (float)stopwatchStartCont.TotalSeconds;

			hoursTransform.localRotation =
				Quaternion.Euler(0f, stopHour * degreesPerHour, 0f);
			minutesTransform.localRotation =
				Quaternion.Euler(0f, stopMinute * degreesPerMinute, 0f);
			secondsTransform.localRotation =
				Quaternion.Euler(0f, stopSecond * degreesPerSecond, 0f);

			UpdateDigital((int)digitalTime.TotalHours, (int)digitalTime.Minutes, (int)digitalTime.Seconds);
		}
	}

	void UpdateDiscrete () {
		DateTime time = DateTime.Now;
		if (!stopwatch)
		{
			hoursTransform.localRotation =
				Quaternion.Euler(0f, time.Hour * degreesPerHour, 0f);
			minutesTransform.localRotation =
				Quaternion.Euler(0f, time.Minute * degreesPerMinute, 0f);
			secondsTransform.localRotation =
				Quaternion.Euler(0f, time.Second * degreesPerSecond, 0f);
			UpdateDigital(time.Hour, time.Minute, time.Second);
		}
		if (stopwatchRunning)
		{
			TimeSpan stopTime = time - stopwatchStartDisc;

			hoursTransform.localRotation =
				Quaternion.Euler(0f, stopTime.Hours * degreesPerHour, 0f);
			minutesTransform.localRotation =
				Quaternion.Euler(0f, stopTime.Minutes * degreesPerMinute, 0f);
			secondsTransform.localRotation =
				Quaternion.Euler(0f, stopTime.Seconds * degreesPerSecond, 0f);

			UpdateDigital((int)stopTime.TotalHours, stopTime.Minutes, stopTime.Seconds);
		}
	}
}