Check for the normality (JQ, QQ plot etc) as Pearson(r) correlation assumes normal distribution ( along with Linearity and Homoscedasticity) and you can get very different estimates if these assumptions are violated. My guess is your data exhibits non normality or heteroskedasticity or both, Kendall rank or Spearman rank correlation should serve your purpose. But do check the data first. You might like to have a look at Chen, P. Y., & Popovich, P. M. (2002). Correlation: Parametric and nonparametric measures. Thousand Oaks, CA: Sage Publications

The Pearson correlation looks for a linear association.  There are other ways data could be associated (quadratic, step function, etc.) without being linear.  Make a scatter plot to see what is going on.

My guess is that you rank-ordered the data 1 (highest), 2, 3, etc. That way the case with the highest raw score has the lowest rank. So the correlations will always be opposite in direction.

I think Stephen may be right. There will usually be a difference in the value of the correlation coefficient between different methods but there shouldn't be  difference in sign. Always plot the data with independent (predictor) variable on the  x axis and the dependent (response) on the y axis. This will show the sign of the correlation. Since there is only one graph there can only be one sign. Of course the relationship may be non linear etc.

Two variables do not usually tell the whole story, so it is possible to get all kinds of weirdness when only comparing X and Y.  Fit other variables in to get the bigger picture.

Thank you all dear professors and colleagues.

After scrutinizing the data again, I found out that it was because of data non-normality.

I appreciate you again.

 Pearson used to determine the relationship between continuous variables , while Spearman is used to determine the relationship between ordinal variables

Ahmed has the real answer, normality is essentially irrelevant until you get to the question of various tests. First of all the correlation coefficient is a second moment characteristic, given a data set you can always compute the sample correlation coefficient but if one of the random variables does not have a second order moment (e.g. variance) then the sample correlation coefficient does not relate to anything.

If you are concerned about normality then the real question is whether X,Y are jointly normal (this is a stronger condition than X,Y separately being normal)l. IF X, Y are jointly normal then rho (not the same as r) is zero if and only if X, Y are independent. If rho is one then X, Y are linearly dependent.

Henry's caution is important, sometimes two variables appear to be correlated but in fact it is because they are separately correlated with a third variable.

rho denotes the theoretical correlation, whereas r is the sample correlation for a given data set

I strongly disagree. Everything you say is correct, but it misses the point. The two correlations coefficients are both strong, but are opposite in direction. That means that one of the variables was scored in the reverse direction of the other. This happens all the time when people use rank-ordered data. They naturally term the highest scorer #1 and count down from there. But the raw scores, just as naturally, count up. It's a simple thing, and there is no error in the data.