The main thing to look at would be what weight they plan on using. We used 30lbs, because of our pitch and the fact that we get snow, ice and frequent storms. But 15lb is generally what's used for standard asphalt shingles. Here's a good link for you.

The correct term is roofing "underlayment" commonly used under asphalt or slate roofing shingles. It can be made of permeable asphalt-saturated cellulose (felt made from recycled cotton rags is no longer available nor is coal tar) or it can be made of impermeable synthetic materials.

In a cold climate the lower part of the roof at the eaves and at the valleys is covered with self-adhering flexible flashing (Grace Ice & Water Shield or its imitators) and the rest is covered with an underlayment. I am in MA and typically from here and to the north the entire roof is covered with I&WS.

Building paper underlayment has several standards and you are not likely to get much asphalt in the material if you specify #15 (no longer called 15# since it doesn't weight 15# per 100 s.f. "square"). A better spec is #30 which is less than 30# but heavier than #15.

There are many superior products available but the the biggest problem with today's asphalt-saturated paper is that it is made from the residue from oil refining and the quality of the asphalt has dropped as oil refining techniques improve to meet the challenge of lower fuel prices. There will inevitably be a point where it is no longer an acceptable material for use as roofing underlayment. IMO that happened many years ago.

If you are comparing bids, the bidders should have been told what the underlayment will be and what quality standards must be met. Leaving it up to the roofer is unwise. In order to control the quality of the project you must control the design shown on the bid documents.

Where is the project located?

Here is a discussion of asphalt-saturated roofing underlayment.

Underlayment is not even required in the Building Code I work under. However, most roofers use some form of it. (I did one roof without it and haven't seen any problems over 12 years.)

When #15 felt is removed, roofers have told me it falls apart like dried leaves. So when I do use underlayment, it is #30. Ice and water shield for the first three feet minimum and in valleys.

More costly synthetic roof underlayments are increasingly popular too. Building Science Corp. says they work as well or better than traditional felt. But some of them are quite slippery and I'd prefer that roofers on my projects don't take any dives.

I haven't specified building paper as a roofing underlayment in 40 years. Not everyone has that budgetary luxury.

OTOH, passivhaus builders sometimes cover the entire exterior sheathing in the stuff.

I don't disagree about cost, I just consider roof waterproofing a necessity rather than a luxury. Its necessary to compromise on the details of a house but some compromises can only be made by the owner since the designer can be held responsible for a failed design and roof leaks can be expensive. Its ironic that the first roof I covered with I&WS was due to the GC's insistence. But we get a lot more freeze-thaw cycles in the winter in New England so the issue is far more critical here.

Here's what $40,000 of interior damage looks like from the outside.

We gave up on tarpaper for roofs years ago when Titanium brand underlayment came out. I high-wind areas we used to need to "strap" the roof to prevent the felt from tearing off. Titanium is woven, reinforced and can't be torn.