There are quite a lot of differences, some more subtle than others, but I tend to find this is the best angle for identification: the side view. Although as we'll see later in this answer, the top/bottom view has a fairly obvious difference, and there's a clue that we can use to identify from the front or rear
So what are you looking at?
On the left is the Hurricane. It's slightly larger than the Spitfire, although that's not always a great help from distance. Perhaps more usefully, the fuselage is fundamentally different shape: I'll elaborate on this in several areas where we can directly compare.
Firstly, the rear fuselage. At first glance the two are very similar, but the Hurricane has a much more pronounced "drop" to the tail, while the Spitfire's is a "rise".
What do I mean by that? Follow the line of the top of the cockpit canopy: the Hurricane's fuselage behind the canopy drops away, while the bottom of the fuselage is relatively flat. On the Spitfire, it's reversed - the top of the fuselage follows the line of the canopy, while the bottom comes up to meet it. It's a subtle difference at first glance, but once you've seen it once it's a very clear distinction. When looking at the Hurricane's rear fuselage, it almost appears to have a "dog leg", where the rearmost area drops in relation to the middle of the aircraft. The Spitfire's follows a much more continuous line
The Spitfire is also clearly much more sleek - it's a "pencil" shaped fuselage, long and thin and curvy. The Hurricane is more blunt and "solid".
Other differences? Look at the tail. The Spitfire's is relatively small and curves upward from the top of the fuselage in a smooth curve, while the Hurricane's comes off at a sharper angle. Similarly the Hurricane's horizontal stabilizer is higher - almost in line with the top of the fuselage.
The Engine cowling shape, as with the cockpit, is different: where again the Spitfire's continues a nearly straight line from the bottom of the cockpit, the Hurricane's has a distinct curve.
If you can get closer, you will also note that the Hurricane's fuselage is almost "ridged" - it's a metal frame with a fabric covering, and that gives it a bumpy appearance, vaguely visible on the above photo.
What about if you can't get a side view, though? Here's the Spitfire (below) and Hurricane (above) from the front... but you can see the same from the rear
Well from the front, there's a single very nice clue under the fuselage: the radiator housing: that box-like structure hanging from the bottom of the plane. The Hurricane has one, and it's on the centre of the fuselage. The Spitfire's is not, it's under the right wing. On some later models, there are two, one under each wing.
The next photo is included mostly for the wing shape, but the aforementioned radiator boxes are visible too: from this, we can tell the Spitfire is a relatively late war model: a Mk IX (9) onward: although there are other subtle clues that suggest this is a much later model... the 5 bladed propeller indicates a Griffon engine, so a Mk XIV (14) or later. Anyway, we can see the Hurricane's single centre radiator, and the Spitfire with one under each wing
And finally, how about the bottom? Well, the Spitfire's wing is extremely distinctive: it has an "Elliptical" wing, shared with only one or two other aircraft. Once you can recognise the Elliptical wing, you can recognise the Spitfire very easily in flight. The Spitfire's wing, particularly the trailing edge, is very curved compared to most other contemporary aircraft. The Hurricane's wings are much more conventionally shaped, with both leading and trailing edges being straight (a trait shared with aircraft like the Typhoon/Tempest, P-51, Me109, A6M "Zero" and most other aircraft from the WW2 era)
Late War Spitfires (the above is a Mk 22) are even more distinctive: they have a "bubble" canopy that protrudes entirely from the fuselage.