Why is propenoic acid more acidic than p-methoxybenzoic acid?
Why is propenoic acid more acidic than p-methoxybenzoic acid? 


This is propenoic acid, $\mathrm{pK_a}=$4.25



This is p-methoxybenzoic acid, $\mathrm{pK_a}=$4.47



An electron withdrawing group increases the acidity of a carboxylic acid. It disperses negative charge by inductive/resonance effect and stabilizes the carboxylate ion. Thus, propenoic acid (pKa 4.25) is more acidic than p-methoxybenzoic acid (pKa 4.47). 

However the molecular structures are totally different. It seems to me to be comparing apples and oranges! To me such hand waving comparisons between molecules are only meaningful if the two molecules are similar.




  The $\mathrm{p}K_\mathrm{a}$ of benzoic acid is $4.20.$ Why does the $\ce{-OMe}$ group make such a big effect on $\mathrm{p}K_\mathrm{a}?$ 


This is a more interesting question. 

\begin{array}{|c|c|} \hline \hline
\text{acid} &amp; \text{pka}  \\ \hline
\text{2-Methoxybenzoic acid} &amp; 4.09   \\ \hline
\text{3-Methoxybenzoic acid} &amp; 4.10  \\ \hline
\text{Benzoic acid} &amp; 4.19  \\ \hline
\text{4-Methoxybenzoic acid} &amp; 4.50  \\ \hline
\end{array}

So the phenyl group has about the same effect on the acidity of a carboxylic acid as observed with an ethylene group. However the substitution of other function groups, and their position on the ring can increase or decrease the pKa of the acid. 

Data sources: 
2-Methoxybenzoic acid


3-Methoxybenzoic acid 


Benzoic acid 


4-Methoxybenzoic acid