Handmade rugs are made in many different places all around the world such as; East, Central and South Asia, Scandinavia, Balkans, Caucasia, North Africa and even in the Americas by the Natives but among all other rugs Turkish and Persian rugs have a different and special status because of their quality, diversity and historical background in rug weaving.
The first difference we need to know between Turkish rugs and Persian rugs is the knotting technique. Turkish rugs are knotted with "double knot" technique also known as "Turkish knot" and Persian rugs are knotted with "single knot" also known as "Persian knot" Please check the photos below to see the two different knotting technique;
Majority of the Turkish rugs are knotted with Turkish knots and it is almost impossible to see a Turkish rug knotted with Persian Knot technique but we can not say the same for Persian rugs, majority of the rugs knotted in Northern and North Western Iran are knotted with Turkish knot technique because even thou they are classified under the name of "Persian rugs" since they are made in the country Iran they are woven by EthnicTurkish people and those ethnic Turkish people use the "Turkish knots" on their rugs. Northern Part of the Iran is inhabited by ethnic Turks for around a thousand year and the region is also called "Southern Azerbaijan" and those Turks in Northern Iran (Southern Azerbaijan) are still keeping their own traditional way of rug making. Except the Northern part rest of the Iran is knotting rugs with Persian knots also other "indo-iranian" countries are using Persian knot to knot their rugs such as Pakistan,India,Afghanistan.
The most obvious difference between a Turkish rug and a Persian rug is their designs. Most of the Persian rugs have more rounded,oriental and elegant designs and motifs, mostly the center of the rug has a medallion design and Persian rugs looks like they are made for a palaces. One of the reasons for that is the single knot technique, because single knot gives the weaver ability to knot more flamboyantly, it is much easier to knot rounded motifs with single knot technique but that is not the only reason, i personally believe that it is to do with the cultural background of the nation, Persian people are one of the earliest settled people group in the world, they had settled in the region for more than two thousand years and they have a long history of imperial states that had control over the neighbouring countries and Persia was one of the superpowers in the world specially in the ancient history, that tradition of civilization and richness influinced the Persian people culturally and they reflect it on their rugs just like they do on other arts.
In the other hand, most of the Turkish rugs have straight, sharp and geometric designs and full of tribal and meaningful motifs. Turkish rugs are like puzzles and mysteries to be solved, sometimes one rug contains so much symbolic motifs that you can write a long story out of the motifs on the rug. One of the reasons why Turkish rugs are made with geometric designs is the double knot technique used on the making of them. It is harder to knot rounded motifs and designs with double knot technique but ofcourse this is not the main reason why Turkish rugs are mostly geometric because we know when it is wished Turkish weavers can also weave perfect oriental, palace style rugs with double knot technique, we can see that on Kayseri, Ladik and especially Hereke and Oushak rugs. They are all Turkish rugs woven with double knot technique but they have amazing elegant looks to the point that Oushak and Hereke rugs are accepted as the world's most quality rugs by many rug experts. Basically the main reason for geometric designs on Turkish rugs are not the double knot technique. Again, i believe the reason for this is the cultural background as it is on Persian rugs. Turkish people are originally from Central Asian and Eurasian steppes and they have been living as nomads for thousand of years and herding large herds. That is probably the reason why Turks have the oldest history of rug making in the world because they lived in tents and they needed textiles to cover the interior parts of the tents, they had wools from their herds and they used them to make rugs and rugs has been an indispensable part of their lives since then. The life was very harsh and difficult in Eurasian steppes for nomadic people and they had adopted to that life style, they became real survivors and fierce warriors and ofcourse their culture was not as "elegant" and soft as the settled people living in cities. That toughness is reflected on Turkish rugs as straight and geometric patterns and we also see many mofits from the nature on Turkish rugs such as trees, rivers and animals and that is because the Turks were living with the nature without the cities seperating from it. It is also very common to see Turkish symbolic signs called "tamga" on Turkish rugs which are mostly to do with the ancient "Tengriism" religion of Turks and tribal signs.
If you like oriental and palace style rugs and want to decorate your place with elegant furnitures then i can say that Persian rugs or Kayseri, Hereke and Oushak Turkish rugs are more suitable for you but if you want your place to look natural, cool, alive and "bohemian" then Turkish rugs would fit your taste the best.