Blood is not typically used as a normal for expression differences. In fact, the most common paradigm is tumor/normal DNA and tumor only RNA. As you imply, expression or methylation profiles will be quite different between RNA from a solid tumor and RNA from the blood.

You can use matched normal blood with tumor RNA for variant calling, as the background mutational profile of SNPs will be preserved in both. The caveat is that you're going to miss mutations in genes that are poorly expressed (which may or may not be fine, depending on your experimental design)

It's also worth noting that adjacent normal tissue is often a poor control anyway, for two reasons: 1) tumor infiltration (or premalignant states) is often observed in nearby tissue, meaning that causative mutations/expression changes will be present in your normal (which is bad). 2) changes in the microenvironment may cause quite different expression profiles than you'd see in a true normal.

That all aside, you often can't collect matched normal tissue - can't just go scooping out a chunk of healthy brain to match up with your GBM tumor. Even if you could collect truly matched normal tissue (from the other breast in BRC, for example), there's the issue of subjecting the patient to another procedure that carries risks and may not provide clear benefit.

Like dariober pointed out, blood is easier to obtain. In fact, if you want to collect a "normal" sample of the exact same tissue, you have to extract material at a location that is distinct to the tumor to make sure that it does not contain any tumor material. This would mean to further destroy tissue of an already damaged organ which is probably not such a good idea.

Regarding differences, if you want, for example, identify the mutations causing the tumor, it is not required to have a sample from the exact same tissue. You have to analyze both samples and ignore all variants found in the normal sample, i.e., mutations that are inherent to the patient and found in all tissues including the blood.

I think part of the reason is that blood is easier to obtain.