Although they're both made from cooked tomatoes, tomato paste has a more intense tomato flavor and thicker texture than tomato puree. These qualities mean that you can't simply substitute paste for puree on a 1-to-1 basis. You may need to add liquid and possibly change seasonings when using paste instead of puree.
For many recipes, you can substitute either 2 to 3 tablespoons of tomato paste plus enough water to make 1 cup for each cup of puree. For example, if your recipe calls for 2 cups of puree, substitute 4 to 6 tablespoons of paste plus enough water added to the paste to equal 2 cups.
Taste the liquid after adding the minimum amount of tomato paste. If it tastes too weak, add 1 or 2 additional tablespoons of paste.
Because tomatoes are cooked only briefly before being made into puree, the tomato flavor in puree is milder than the flavor of paste. The concentrated tomato flavor of paste comes from tomatoes that are cooked for hours until they're reduced to the paste consistency. Use a small amount of tomato paste if you want a closer match to the mild flavor of puree.
For additional flavor in sauces, meat loaf toppings or other dishes, use chicken or beef bouillon for half of the water you use when substituting tomato paste for puree.
Tomato paste has no added salt and naturally contains about 20 mg. of salt in a 2-tablespoon serving. On the other hand, tomato puree generally contains added salt; it typically contains 62 mg. of salt for the same serving size. If you're using paste instead of puree, however, you may not need additional salt. Taste the tomato liquid before adding any additional salt.