I am comparing two ordinal variables (i.e., two independent Likert items). I've used a chi-squared test to test significance of a relationship between the two, and a Pearson correlation (I'm expecting the two variables to have a linear relationship) to test the strength. I have a sample size of 250 and I'm looking at a table running 4x5 (one ordinal item has four integers, and the other has 5). Admittedly, not all cells in the contingency table have a value of at least 5. After running the tests, I got a very significant p-value for the chi-squared (~10^-7), but the correlation coefficient is quite low (~0.2). Can someone explain to me how this makes sense? Or am I using the wrong methods? What would be a better method to use?
Erring on the side of being pedagogical, Pearson correlation is not recommended for ordinal variables. Even after considering the Likert scales aspect of your data I would still be wary of using Pearson's because of the number of assumptions it requires.
Investigate Spearman or Kendall's correlations instead for effects like whether or not both measures are given by every person i.e. paired etc.
Another similar question that might be useful.
My first question is: What is the difference between using a Chi-squared test vs the Spearman Rho's test?
The chi-squared test treats both variables $X$ and $Y$ as nominal (like colors, countries etc.) and thus, it can detect any sort of underlying relationship between $X$ and $Y$. In contrast, tests for linear resp. rank-correlations make use of the fact that the factor levels are ordered and are particularly suitable to detect linear resp. monotone underlying relationships. In your setting, it seems to make sense to condense the relationship to a correlation coefficient, so it would be more natural to provide p values associated with that measure instead of the less focused chi-squared test. But basically it is up to you.
Of course you can't just run all tests that you know and then pick the one with the smallest p value. Ideally, you already select an analysis strategy before looking at the data to avoid data snooping and to end up with reproducible conclusions.
PS: You are also free to use linear correlations instead of rank-correlations. Its test is basically the "linear-by-linear" test for association by Agresti, one of the godfathers of modern categorical data analysis. If you are interested, his famous book  is worth every penny. You will find it in every university library.
And my second question is: Why are the correlation coefficients so low when the Chi-squared test looks significant?
A small p value means strong evidence against the null hypothesis "no relationship between $X$ and $Y$". Depending on the sample size, a sample correlation of 0.2 can mean extremely strong evidence or, if the sample is small, not much evidence against this null hypothesis. Or in other words: the p value is not a measure of effect size.
 Agresti, A. (2002). Categorical Data Analysis, Second Edition. Hoboken, New Jersey: John Wiley & Sons.