Can someone tell me the difference?
Schauen Sie sich dieses Foto an.
Schauen Sie dieses Foto an.
Are they both correct? Do they both have the same meaning?
They are both correct. Both are very similar, and in many interchangable. Yet they have a difference. "Sich etwas anschauen" to look/watch/view something in a more concentrated way. You try to check the picture for its details. "Etwas anschauen" to look/watch/view something but you focus less on the details.
When you ask your art professor to judge your photo, then you ask/tell him to "Schauen Sie sich das Foto an.". But when you want someone to just look at the photo, then you just "Schauen Sie das Foto". Your photo is hanging in a gallery, but half of the picture is not really visible because the lighting is terrible. Youre telling the janitor "Schauen Sie das Foto an, (ist da irgendwas falsch?)" (Look at the picture, is there anything wrong). You hope that the janitor notices, that half of the picture is not enjoyable because its to dark (or for whatever reason).
Other example. A police officer shows a picture to two different persons. The police officer shows the picture to a barista and asks him to look at the picture and if he ever saw this girl. "Schauen Sie sich das Bild an. Haben Sie dieses Mädchen schon mal gesehen?". The officer wants the barista to closely look at the girl, so that he can answer if he saw or saw not this girl. The police officer interrogates a suspicious person. The suspicious person does not say a word. The officer shows pictures of the dead victim and tells the suspicious person "Schau das Bild an. Das ist deine Schuld!". The officer wants him to see the scary leftovers of the victim, so that the suspicious person gets emotional. Here it does not matter how the details look, the overall of the picture shall be seen and you do not need to notice that just 3 and not 4 fingers were cut off.
Schau dir das an...i like saying "check it out" in English. Otherwise, just take a look.
Adding the dative or reflexive, depending on the verb, is similar to saying "for yourself," which is also not necessarily stated in English. To me both emphasize for one's own benefit. One is more neutral.