There are several important differences between the logics of Aristotle and Frege.
1.Aristotle understood sentences to be fundamentally of the form Subject-Predicate. He classified sentences into the figures (A) All S is P; (E) All S is not P; (I) Some S is P; (O) Some S is not P. This is very limiting. If we want to say "John loves Mary", is 'John' the subject and 'loves Mary' is being predicated of John? or is 'Mary' really the subject and 'John loves' is being predicated of Mary? It seems more reasonable to say that 'loves' is the predicate term on its own and that 'loves' is being predicated of John and Mary together.
One of Frege's insights is that the logical form of sentences is not fundamentally subject-predicate, but rather more like mathematical functions. If we define a function SQR(x, y) to be true when x is the square of y and false otherwise, then we might say that the values x=4/y=2 satisfy the function SQR, but the values x=4/y=3 do not. In the same way we can understand the sentence "John loves Mary" as a two place predicate Loves(x, y) which is satisfied by x=john/y=mary. Nor are we limited to just two places. 'Between(x, y, z)' might be a three place predicate satisfied by x=alice/y=bob/z=charlie just in case the sentence "Alice is between Bob and Charlie" is true. So whereas for Aristotle a predicate is a property of a particular or universal subject, for Frege it is an n-place function with variables such that it is true when satisfied by certain substitution instances of those variables.
2.The difference is broader than just the approach to predicates. Frege borrowed from Boole and de Morgan the idea that propositions can be considered as variables that can have the values true or false. We accept this as a commonplace today, but it was revolutionary at the time. For Aristotle, logic is about stating propositions that are true and then proving other true things from them. For Frege, propositions may be used as the premises of an argument without our being committed to whether they are true or not. Frege distinguished between 'thought' and 'judgement', which in more modern terminology we might call 'proposition' and 'assertion'. We can distinguish between the thought of "Alice's being taller than Bob" and the judgement that "Alice is taller than Bob". We might observe that "Alice's being taller than Bob" together with "Bob's being taller than Charlie" entails "Alice's being taller than Charlie", but this does not commit us to claiming that any of these propositions is actually true.
Once we have propositions as boolean variables, we can make use of the boolean logic of connectives (and, or, not, etc.), which again is completely lacking in Aristotle.
3.Another difference is that for Aristotle the quantifiers 'all' and 'some' only appear once within a sentence. In Frege's logic, the quantifiers can be combined to express more complex propositions. For example, it allows us to express the difference between sentences such as
(∀x)(∃y)(Boy(x) → (Girl(y) ∧ Loves(x, y)) - Every boy loves some girl; and
(∃y)(∀x)(Girl(y) ∧ (Boy(x) → Loves(x, y)) - There is some girl whom every boy loves.
Frege's logic even allows us to prove that the second sentence entails the first. These sentences cannot be written using Aristotle's logic.
A noteworthy corollary of Aristotle's approach is that it takes the sentence "all S is P" as having existential import, i.e. it assumes that there are some S. Aristotle has no use for the sentence "all sheep are mammals" if there are no sheep. By contrast, Frege's logic takes the universal quantifier 'all' to be hypothetical, so a sentence of the form "all S is P" might be glossed as "anything that is S is also P". This is highly useful, but it does have the unintuitive consequence that the sentences "all unicorns are white" and "all unicorns are not white" are both true, because there are no unicorns.