Do men and women have the same number of genes?
It is true that the Y chromosome is shorter than the X chromosome and that there are more genes on the X chromosome.


  Do men have fewer genes?


One could (mis)understand three things in the expression "number of genes". 


Number of gene copies (see Copy Number Variation)
Number of genes
Number of alleles


Thanks to @GerardoFurtado for correcting my semantic in the comments.

1. Number of gene copies

From the statement that there are fewer genes on the Y chromosome, one can conclude that men have fewer genes copies than woman.

This is the intuition the OP seemed to have.

2. Number of genes

Men also have an X chromosome. So men have the standard genes present on the X chromosome (but they only have a single copy of it while women have two copies; btw you might be interested in dosage compensation).

Because women do not have a Y chromosome and because there are a number of genes on the Y chromosome that are not present on the X chromosome, men have genes that female don't have at all. Therefore women have fewer genes than men.

3. Number of alleles

There is not much reason to expect that one gender would be more heterozygote than the other at autosomes (=non sexual chromosomes). Some may hypothesize that women may have more heterozygosity than men if there is stronger selection among sperm than among ovules or things like that but let's not get down this complicated path.

One one hand women have more gene copies and therefore might experience more heterozygosity, one the other hand, men have more genes and would therefore eventually carry more alleles. I don't know which side wins!

Did you mean number of genes per cell or per individual?

So far I assumed you were interested about the number of genes (or gene copies) per cells but if you want to compare whole individuals than it is a different story!

Men are on average taller and therefore have more cells. Therefore if you compare the body-wide number of gene copies, women will have fewer gene copy on average (Thanks to @JM97 comment).