The difference between the two arises because Maxwell's equations, while looking perfectly "equal", actually are not all of the same nature when we phrase electromagnetism in terms of a potential. If you think of $F$ as the dynamical variable, then $$ \mathrm{d}F = 0 \quad \mathrm{d}{\star}F = 0$$ in vacuum look perfectly symmetric, and you might imagine adding electric and magnetic current 3-densities (these are the Hodge duals of the standard 1-vector current densitites) $j_\text{el},j_\text{mag}$ to obtain $$ \mathrm{d}F = j_\text{mag}\quad \mathrm{d}{\star}F = j_\text{el}.$$ These would be the "Maxwell equations" of electromagnetism with both magnetic and electric charges. However, this theory has a "problem" - it is rather difficult to write it as a least action formulation. There is one, due to Zwanziger in "Local-Lagrangian Quantum Field Theory of Electric and Magnetic Charges", see also this answer of mine, but it is rather unwieldy and unnatural, and it has to artifically double the d.o.f. by introducing both an electric and a magnetic potential and imposing their field strengths being Hodge dual at the level of the equations of motion. My linked answer also remarks that there is a way to get magnetic monopoles that are not topological in the way you are thinking of here - this question seems to be about the singular Dirac monopoles rather than the non-singular 't Hooft-Polyakov monopoles.

Much more natural is to have the magnetic charges vanish, i.e. $\mathrm{d}F = 0$. Then, locally, by the Poincaré lemma there exists a 1-form potential $A$ with $\mathrm{d}A = F$, and there is the rather natural Yang-Mills Lagrangian with $A$ coupled to a current yielding $\mathrm{d}{\star}F = j_\text{el}$ when $A$ is considered as the dynamical variable. The crucial observation is that in this Lagrangian formulation, $\mathrm{d}F = 0$ is not an equation of motion. It is the Bianchi identity simply following from defining $F$ to be the derivative of the potential $A$, and it is therefore impossible to couple the theory of the electric potential $A$ to a magnetic current. As you already mention, introducing magnetic monopoles into this gauge theory requires "topological trickery", where we have to exclude the position of the monopole from the spacetime we are considering in order to rescue $\mathrm{d}F = 0$ and thus the description in terms of $A$, see also this answer of mine.

Now, you might say that since we introduced $A$ based on Maxwell's equations and these are perfectly symmetric, there is nothing fundamental about the magnetic charge that makes it the one that is supposed to be described in this topological way instead of the electric charge. We can switch $F$ and ${\star}F$, i.e. change which we see as the fundamental quantity and which as the Hodge dual, and define instead the "default" state of our gauge theory to be one in which electric charges are absent, so that we have a magnetic potential $B$ with $\mathrm{d}\mathrm{d}B = \mathrm{d}{\star}F = 0$.

But since electric charges are so plentiful in our everyday world while magnetic ones are not, this is terribly inefficient.

The difference between the two arises because Maxwell's equations, while looking perfectly "equal", actually are not all of the same nature when we phrase electromagnetism in terms of a potential. If you think of $F$ as the dynamical variable, then $$ \mathrm{d}F = 0 \quad \mathrm{d}{\star}F = 0$$ in vacuum look perfectly symmetric, and you might imagine adding electric and magnetic current 3-densities (these are the Hodge duals of the standard 1-vector current densitites) $j_\text{el},j_\text{mag}$ to obtain $$ \mathrm{d}F = j_\text{mag}\quad \mathrm{d}{\star}F = j_\text{el}.$$ These would be the "Maxwell equations" of electromagnetism with both magnetic and electric charges. However, this theory has a "problem" - it is rather difficult to write it as a least action formulation. There is one, due to Zwanziger in "Local-Lagrangian Quantum Field Theory of Electric and Magnetic Charges", see also this answer of mine, but it is rather unwieldy and unnatural, and it has to artifically double the d.o.f. by introducing both an electric and a magnetic potential and imposing their field strengths being Hodge dual at the level of the equations of motion. My linked answer also remarks that there is a way to get magnetic monopoles that are not topological in the way you are thinking of here - this question seems to be about the singular Dirac monopoles rather than the non-singular 't Hooft-Polyakov monopoles.

Much more natural is to have the magnetic charges vanish, i.e. $\mathrm{d}F = 0$. Then, locally, by the Poincaré lemma there exists a 1-form potential $A$ with $\mathrm{d}A = F$, and there is the rather natural Yang-Mills Lagrangian with $A$ coupled to a current yielding $\mathrm{d}{\star}F = j_\text{el}$ when $A$ is considered as the dynamical variable. The crucial observation is that in this Lagrangian formulation, $\mathrm{d}F = 0$ is not an equation of motion. It is the Bianchi identity simply following from defining $F$ to be the derivative of the potential $A$, and it is therefore impossible to couple the theory of the electric potential $A$ to a magnetic current. As you already mention, introducing magnetic monopoles into this gauge theory requires "topological trickery", where we have to exclude the position of the monopole from the spacetime we are considering in order to rescue $\mathrm{d}F = 0$ and thus the description in terms of $A$, see also this answer of mine.

Now, you might say that since we introduced $A$ based on Maxwell's equations and these are perfectly symmetric, there is nothing fundamental about the magnetic charge that makes it the one that is supposed to be described in this topological way instead of the electric charge. We can switch $F$ and ${\star}F$, i.e. change which we see as the fundamental quantity and which as the Hodge dual, and define instead the "default" state of our gauge theory to be one in which electric charges are absent, so that we have a magnetic potential $B$ with $\mathrm{d}\mathrm{d}B = \mathrm{d}{\star}F = 0$.

But since electric charges are so plentiful in our everyday world while magnetic ones are not, this is terribly inefficient.

The difference between the two arises because Maxwell's equation, while looking perfectly "equal", actually are not all of the same nature when we phrase electromagnetism in terms of a potential. If you think of $F$ as the dynamical variable, then $$ \mathrm{d}F = 0 \quad \mathrm{d}{\star}F = 0$$ in vacuum look perfectly symmetric, and you might imagine adding electric and magnetic current 3-densities (these are the Hodge duals of the standard 1-vector current densitites) $j_\text{el},j_\text{mag}$ to obtain $$ \mathrm{d}F = j_\text{mag}\quad \mathrm{d}{\star}F = j_\text{el}.$$ These would be the "Maxwell equations" of electromagnetism with both magnetic and electric charges. However, this theory has a "problem" - it is rather difficult to write it as a least action formulation. There is one, due to Zwanziger in "Local-Lagrangian Quantum Field Theory of Electric and Magnetic Charges", see also this answer of mine, but it is rather unwieldy and unnatural, and it has to artifically double the d.o.f. by introducing both an electric and a magnetic potential and imposing their field strengths being Hodge dual at the level of the equations of motion. My linked answer also remarks that there is a way to get magnetic monopoles that are not topological in the way you are thinking of here - this question seems to be about the singular Dirac monopoles rather than the non-singular 't Hooft-Polyakov monopoles.

Much more natural is to have the magnetic charges vanish, i.e. $\mathrm{d}F = 0$. Then, locally, by the Poincaré there exists a 1-form potential $A$ with $\mathrm{d}A = F$, and there is the rather natural Yang-Mills Lagrangian with $A$ coupled to a current yielding $\mathrm{d}{\star}F = j_\text{el}$ when $A$ is considered as the dynamical variable. The crucial observation is that in this Lagrangian formulation, $\mathrm{d}F = 0$ is not an equation of motion. It is the Bianchi identity simply following from defining $F$ to be the derivative of the potential $A$, and it is therefore impossible to couple the theory of the electric potential $A$ to a magnetic current. As you already mention, introducing magnetic monopoles into this gauge theory requires "topological trickery", where we have to exclude the position of the monopole from the spacetime we are considering in order to rescue $\mathrm{d}F = 0$ and thus the description in terms of $A$, see also this answer of mine.

Now, you might say that since we introduced $A$ based on Maxwell's equations and these are perfectly symmetric, there is nothing fundamental about the magnetic charge that makes it the one that is supposed to be described in this topological way instead of the electric charge. We can switch $F$ and ${\star}F$, i.e. change which we see as the fundamental quantity and which as the Hodge dual, and define instead the "default" state of our gauge theory to be one in which electric charges are absent, so that we have a magnetic potential $B$ with $\mathrm{d}\mathrm{d}B = \mathrm{d}{\star}F = 0$.

But since electric charges are so plentiful in our everyday world while magnetic ones are not, this is terribly inefficient.

The difference between the two arises because Maxwell's equations, while looking perfectly "equal", actually are not all of the same nature when we phrase electromagnetism in terms of a potential. If you think of $F$ as the dynamical variable, then $$ \mathrm{d}F = 0 \quad \mathrm{d}{\star}F = 0$$ in vacuum look perfectly symmetric, and you might imagine adding electric and magnetic current 3-densities (these are the Hodge duals of the standard 1-vector current densitites) $j_\text{el},j_\text{mag}$ to obtain $$ \mathrm{d}F = j_\text{mag}\quad \mathrm{d}{\star}F = j_\text{el}.$$ These would be the "Maxwell equations" of electromagnetism with both magnetic and electric charges. However, this theory has a "problem" - it is rather difficult to write it as a least action formulation. There is one, due to Zwanziger in "Local-Lagrangian Quantum Field Theory of Electric and Magnetic Charges", see also this answer of mine, but it is rather unwieldy and unnatural, and it has to artifically double the d.o.f. by introducing both an electric and a magnetic potential and imposing their field strengths being Hodge dual at the level of the equations of motion. My linked answer also remarks that there is a way to get magnetic monopoles that are not topological in the way you are thinking of here - this question seems to be about the singular Dirac monopoles rather than the non-singular 't Hooft-Polyakov monopoles.

Much more natural is to have the magnetic charges vanish, i.e. $\mathrm{d}F = 0$. Then, locally, by the Poincaré lemma there exists a 1-form potential $A$ with $\mathrm{d}A = F$, and there is the rather natural Yang-Mills Lagrangian with $A$ coupled to a current yielding $\mathrm{d}{\star}F = j_\text{el}$ when $A$ is considered as the dynamical variable. The crucial observation is that in this Lagrangian formulation, $\mathrm{d}F = 0$ is not an equation of motion. It is the Bianchi identity simply following from defining $F$ to be the derivative of the potential $A$, and it is therefore impossible to couple the theory of the electric potential $A$ to a magnetic current. As you already mention, introducing magnetic monopoles into this gauge theory requires "topological trickery", where we have to exclude the position of the monopole from the spacetime we are considering in order to rescue $\mathrm{d}F = 0$ and thus the description in terms of $A$, see also this answer of mine.

Now, you might say that since we introduced $A$ based on Maxwell's equations and these are perfectly symmetric, there is nothing fundamental about the magnetic charge that makes it the one that is supposed to be described in this topological way instead of the electric charge. We can switch $F$ and ${\star}F$, i.e. change which we see as the fundamental quantity and which as the Hodge dual, and define instead the "default" state of our gauge theory to be one in which electric charges are absent, so that we have a magnetic potential $B$ with $\mathrm{d}\mathrm{d}B = \mathrm{d}{\star}F = 0$.

But since electric charges are so plentiful in our everyday world while magnetic ones are not, this is terribly inefficient.